Functions that compute some of the sequences in Sloane’s tables

EXAMPLES:

Type sloane.[tab] to see a list of the sequences that are defined.

sage: a = sloane.A000005; a
 The integer sequence tau(n), which is the number of divisors of n.
 sage: a(1)
 1
 sage: a(6)
 4
 sage: a(100)
 9

Type d._eval?? to see how the function that computes an individual term of the sequence is implemented.

The input must be a positive integer:

sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

You can also change how a sequence prints:

sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a.rename('(..., tau(n), ...)')
sage: a
(..., tau(n), ...)
sage: a.reset_name()
sage: a
The integer sequence tau(n), which is the number of divisors of n.

TESTS:

sage: a = sloane.A000001;
sage: a == loads(dumps(a))
True

AUTHORS:

  • William Stein: framework
  • Jaap Spies: most sequences
  • Nick Alexander: updated framework
class sage.combinat.sloane_functions.A000001
__init__()

Number of groups of order n.

Note: The database_gap-4.4.9 must be installed for n > 50.

run sage -i database_gap-4.4.9 or higher first.

INPUT:

  • n - positive integer

OUTPUT: integer

EXAMPLES:

sage: a = sloane.A000001;a
Number of groups of order n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1) #optional database_gap
1
sage: a(2) #optional database_gap
1
sage: a(9) #optional database_gap
2
sage: a.list(16) #optional database_gap
[1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
sage: a(60)     # optional
13

AUTHORS:

  • Jaap Spies (2007-02-04)
_eval(n)

EXAMPLES:

sage: sloane.A000001._eval(4)
2
sage: sloane.A000001._eval(51) #optional requires database_gap
_repr_()

EXAMPLES:

sage: sloane.A000001._repr_()
'Number of groups of order n.'
class sage.combinat.sloane_functions.A000004
__init__()

The zero sequence.

INPUT:

  • n - non negative integer

OUTPUT:

EXAMPLES:

sage: a = sloane.A000004; a
The zero sequence.
sage: a(1)
0
sage: a(2007)
0
sage: a.list(12)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

AUTHORS:

  • Jaap Spies (2006-12-10)
_eval(n)

EXAMPLES:

sage: sloane.A000004._eval(5)
0
_repr_()

EXAMPLES:

sage: sloane.A000004._repr_()
'The zero sequence.'
class sage.combinat.sloane_functions.A000005
__init__()

The sequence tau(n), which is the number of divisors of n.

This sequence is also denoted d(n) (also called \tau(n) or \sigma_0(n)), the number of divisors of n.

INPUT:

  • n - positive integer

OUTPUT:

EXAMPLES:

sage: d = sloane.A000005; d
The integer sequence tau(n), which is the number of divisors of n.
sage: d(1)
1
sage: d(6)
4
sage: d(51)
4
sage: d(100)
9
sage: d(0)
...
ValueError: input n (=0) must be a positive integer
sage: d.list(10)
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

AUTHORS:

  • Jaap Spies (2006-12-10)
  • William Stein (2007-01-08)
_eval(n)

EXAMPLES:

sage: sloane.A000005._eval(5)
2
_repr_()

EXAMPLES:

sage: sloane.A000005._repr_()
'The integer sequence tau(n), which is the number of divisors of n.'
class sage.combinat.sloane_functions.A000007
__init__()

The characteristic function of 0: a(n) = 0^n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000007;a
The characteristic function of 0: a(n) = 0^n.
sage: a(0)
1
sage: a(2)
0
sage: a(12)
0
sage: a.list(12)
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

AUTHORS:

  • Jaap Spies (2007-01-12)
_eval(n)

EXAMPLES:

sage: [sloane.A000007._eval(n) for n in range(10)]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
_repr_()

EXAMPLES:

sage: sloane.A000007._repr_()
'The characteristic function of 0: a(n) = 0^n.'
class sage.combinat.sloane_functions.A000008
__init__()

Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000008;a
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
16
sage: a.list(14)
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]       

AUTHOR:

    1. Gaski (2009-05-29)
_eval(n)

EXAMPLES:

sage: [sloane.A000008._eval(n) for n in range(14)]
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]
_repr_()

EXAMPLES:

sage: sloane.A000008._repr_()
'Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.'
class sage.combinat.sloane_functions.A000009
__init__()

Number of partitions of n into odd parts.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000009;a
Number of partitions of n into odd parts.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
18
sage: a.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]       

AUTHOR:

  • Jaap Spies (2007-01-30)
_eval(n)

EXAMPLES:

sage: [sloane.A000009._eval(i) for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
_precompute(how_many=50)

EXAMPLES:

sage: initial = len(sloane.A000009._b)
sage: sloane.A000009._precompute(10)
sage: len(sloane.A000009._b) - initial == 10
True
_repr_()

EXAMPLES:

sage: sloane.A000009._repr_()
'Number of partitions of n into odd parts.'
cf()

EXAMPLES:

sage: it = sloane.A000009.cf()
sage: [it.next() for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
list(n)

EXAMPLES:

sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
class sage.combinat.sloane_functions.A000010
__init__()

The integer sequence A000010 is Euler’s totient function.

Number of positive integers i < n that are relative prime to n. Number of totatives of n.

Euler totient function \phi(n): count numbers n and prime to n. euler_phi is a standard Sage function implemented in PARI

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000010; a
Euler's totient function
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(11)
10
sage: a.list(12)
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-12)
_eval(n)

EXAMPLES:

sage: [sloane.A000010._eval(n) for n in range(1,11)]
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4]
_repr_()

EXAMPLES:

sage: sloane.A000010._repr_()
"Euler's totient function"
class sage.combinat.sloane_functions.A000012
__init__()

The all 1’s sequence.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000012; a
The all 1's sequence.
sage: a(1)
1
sage: a(2007)
1
sage: a.list(12)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

AUTHORS:

  • Jaap Spies (2007-01-12)
_eval(n)

EXAMPLES:

sage: [sloane.A000012._eval(n) for n in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
_repr_()

EXAMPLES:

sage: sloane.A000012._repr_()
"The all 1's sequence."
class sage.combinat.sloane_functions.A000015
__init__()

Smallest prime power \geq n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000015; a
Smallest prime power >= n.
sage: a(1)
1
sage: a(8)
8
sage: a(305)
307
sage: a(-4)
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000015._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11]
_repr_()

EXAMPLES:

sage: sloane.A000015._repr_()
'Smallest prime power >= n.'
class sage.combinat.sloane_functions.A000016
__init__()

Sloane’s A000016

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000016; a
Sloane's A000016.
sage: a(1)
1
sage: a(0)
1
sage: a(8)
16
sage: a(75)
251859545753048193000
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000016._eval(n) for n in range(10)]
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30]
_repr_()

EXAMPLES:

sage: sloane.A000016._repr_()
"Sloane's A000016."
class sage.combinat.sloane_functions.A000027
__init__()

The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

The following examples are tests of SloaneSequence more than A000027.

EXAMPLES:

sage: s = sloane.A000027; s
The natural numbers.
sage: s(10)
10

Index n is interpreted as _eval(n):

sage: s[10]
10

Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term:

sage: s[:3]
[1, 2]
sage: s[3:6]
[3, 4, 5]
sage: s.list(5)
[1, 2, 3, 4, 5]
_eval(n)

EXAMPLES:

sage: sloane.A000027._eval(5)
5
_repr_()

EXAMPLES:

sage: sloane.A000027._repr_()
'The natural numbers.'
class sage.combinat.sloane_functions.A000030
__init__()

Initial digit of n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000030; a
Initial digit of n
sage: a(0)
0
sage: a(1)
1
sage: a(8)
8
sage: a(454)
4
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000030._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
_repr_()

EXAMPLES:

sage: sloane.A000030._repr_()
'Initial digit of n'
class sage.combinat.sloane_functions.A000032
__init__()

Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000032; a
Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
sage: a(0)
2
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000032._eval(n) for n in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
_repr_()

EXAMPLES:

sage: sloane.A000032._repr_()
'Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).'
class sage.combinat.sloane_functions.A000035
__init__()

A simple periodic sequence.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000035;a
A simple periodic sequence.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
0
sage: a(9)
1
sage: a.list(10)
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]       

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A000035._eval(n) for n in range(10)]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
_repr_()

EXAMPLES:

sage: sloane.A000035._repr_()
'A simple periodic sequence.'
class sage.combinat.sloane_functions.A000040
__init__()

The prime numbers.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000040; a
The prime numbers.
sage: a(1)
2
sage: a(8)
19
sage: a(305)
2011
sage: a.list(12)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-17)
_eval(n)

EXAMPLES:

sage: [sloane.A000040._eval(n) for n in range(1,11)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
_repr_()

EXAMPLES:

sage: sloane.A000040._repr_()
'The prime numbers.'
class sage.combinat.sloane_functions.A000041
__init__()

a(n) = number of partitions of n (the partition numbers).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000041;a
a(n) = number of partitions of n (the partition numbers).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
22
sage: a(200)
3972999029388
sage: a.list(9)
[1, 1, 2, 3, 5, 7, 11, 15, 22]

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000041._eval(n) for n in range(1,11)]
[1, 2, 3, 5, 7, 11, 15, 22, 30, 42]
_repr_()

EXAMPLES:

sage: sloane.A000041._repr_()
'a(n) = number of partitions of n (the partition numbers).'
class sage.combinat.sloane_functions.A000043
__init__()

Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000043;a
Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
sage: a(1)
2
sage: a(2)
3
sage: a(39)
13466917
sage: a(40)
...
IndexError: list index out of range
sage: a.list(12)
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000043._eval(n) for n in range(1,11)]
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89]
_repr_()

EXAMPLES:

sage: sloane.A000043._repr_()
'Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.'
class sage.combinat.sloane_functions.A000045
__init__()

Sequence of Fibonacci numbers, offset 0,4.

REFERENCES:

We have one more. Our first Fibonacci number is 0.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000045; a
Fibonacci numbers with index n >= 0
sage: a(0)
0
sage: a(1)
1
sage: a.list(12)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A000045._eval(n) for n in range(1,11)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
_precompute(how_many=500)

EXAMPLES:

sage: initial = len(sloane.A000045._b)
sage: sloane.A000045._precompute(10)
sage: len(sloane.A000045._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A000045._repr_()
'Fibonacci numbers with index n >= 0'
fib()

Returns a generator over all Fibonacci numbers, starting with 0.

EXAMPLES:

sage: it = sloane.A000045.fib()
sage: [it.next() for i in range(10)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
list(n)

EXAMPLES:

sage: sloane.A000045.list(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
class sage.combinat.sloane_functions.A000069
__init__()

Odious numbers: odd number of 1’s in binary expansion.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000069; a
Odious numbers: odd number of 1's in binary expansion.
sage: a(0)
1
sage: a(2)
4
sage: a.list(9)
[1, 2, 4, 7, 8, 11, 13, 14, 16]

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A000069._eval(n) for n in range(10)]
[1, 2, 4, 7, 8, 11, 13, 14, 16, 19]
_repr_()

EXAMPLES:

sage: sloane.A000069._repr_()
"Odious numbers: odd number of 1's in binary expansion."
class sage.combinat.sloane_functions.A000073
__init__()

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000073;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(11)
149
sage: a.list(12)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

AUTHORS:

  • Jaap Spies (2007-01-19)
_eval(n)

EXAMPLES:

sage: [sloane.A000073._eval(n) for n in range(10)]
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
_precompute(how_many=20)

EXAMPLES:

sage: initial = len(sloane.A000073._b)
sage: sloane.A000073._precompute(10)
sage: len(sloane.A000073._b) - initial == 10
True
_repr_()

EXAMPLES:

sage: sloane.A000073._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
list(n)

EXAMPLES:

sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
class sage.combinat.sloane_functions.A000079
__init__()

Powers of 2: a(n) = 2^n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000079;a
Powers of 2: a(n) = 2^n.
sage: a(0)
1
sage: a(2)
4
sage: a(8)
256
sage: a(100)
1267650600228229401496703205376
sage: a.list(9)
[1, 2, 4, 8, 16, 32, 64, 128, 256]

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000079._eval(n) for n in range(10)]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
_repr_()

EXAMPLES:

sage: sloane.A000079._repr_()
'Powers of 2: a(n) = 2^n.'
class sage.combinat.sloane_functions.A000085
__init__()

Number of self-inverse permutations on n letters, also known as involutions; number of Young tableaux with n cells.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000085;a
Number of self-inverse permutations on n letters.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
140152
sage: a.list(13)
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

AUTHORS:

  • Jaap Spies (2007-02-03)
_eval(n)

EXAMPLES:

sage: [sloane.A000085._eval(n) for n in range(10)]
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620]
_repr_()

EXAMPLES:

sage: sloane.A000085._repr_()
'Number of self-inverse permutations on n letters.'
class sage.combinat.sloane_functions.A000100
__init__()

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000100;a
Number of compositions of n in which the maximum part size is 3.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
0
sage: a(3)
1
sage: a(11)
360
sage: a.list(12)
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000100._eval(n) for n in range(10)]
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94]
_repr_()

EXAMPLES:

sage: sloane.A000100._repr_()
'Number of compositions of n in which the maximum part size is 3.'
class sage.combinat.sloane_functions.A000108
__init__()

Catalan numbers: C_n = \frac{{{2n}\choose{n}}}{n+1} = \frac {(2n)!}{n!(n+1)!}. Also called Segner numbers.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000108;a
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
1430
sage: a(40)
2622127042276492108820
sage: a.list(9)
[1, 1, 2, 5, 14, 42, 132, 429, 1430]

AUTHORS:

  • Jaap Spies (2007-01-12)
_eval(n)

EXAMPLES:

sage: [sloane.A000108._eval(n) for n in range(10)]
[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
_repr_()

EXAMPLES:

sage: sloane.A000108._repr_()
'Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.'
class sage.combinat.sloane_functions.A000110
__init__()

The sequence of Bell numbers.

The Bell number B_n counts the number of ways to put n distinguishable things into indistinguishable boxes such that no box is empty.

Let S(n, k) denote the Stirling number of the second kind. Then

B_n = \sum{k=0}^{n} S(n, k) .

INPUT:

  • n - integer = 0

OUTPUT:

  • integer - B_n

EXAMPLES:

sage: a = sloane.A000110; a
Sequence of Bell numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
sage: a.list(10)
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

AUTHORS:

  • Nick Alexander
_repr_()

EXAMPLES:

sage: sloane.A000110._repr_()
'Sequence of Bell numbers'
class sage.combinat.sloane_functions.A000120
__init__()

1’s-counting sequence: number of 1’s in binary expansion of n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000120;a
1's-counting sequence: number of 1's in binary expansion of n.
sage: a(0)
0
sage: a(2)
1
sage: a(12)
2
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000120._eval(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
_repr_()

EXAMPLES:

sage: sloane.A000120._repr_()
"1's-counting sequence: number of 1's in binary expansion of n."
f(n)

EXAMPLES:

sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
class sage.combinat.sloane_functions.A000124
__init__()

Central polygonal numbers (the Lazy Caterer’s sequence): n(n+1)/2 + 1.

Or, maximal number of pieces formed when slicing a pancake with n cuts.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000124;a
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
4
sage: a(9)
46
sage: a.list(10)
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000124._eval(n) for n in range(10)]
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]
_repr_()

EXAMPLES:

sage: sloane.A000124._repr_()
"Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1."
class sage.combinat.sloane_functions.A000129
__init__()

Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2a(n-1) + a(n-2).

Denominators of continued fraction convergents to \sqrt 2.

See also A001333

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000129;a
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
sage: a(0)
0
sage: a(2)
2
sage: a(12)
13860
sage: a.list(12)
[0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

AUTHORS:

  • Jaap Spies (2007-01-25)
_repr_()

EXAMPLES:

sage: sloane.A000129._repr_()
'Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A000142
__init__()

Factorial numbers: n! = 1 \cdot 2 \cdot 3 \cdots n

Order of symmetric group S_n, number of permutations of n letters.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000142;a
Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
sage: a(0)
1
sage: a(8)
40320
sage: a(40)
815915283247897734345611269596115894272000000000
sage: a.list(9)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320]       

AUTHORS:

  • Jaap Spies (2007-01-12)
_eval(n)

EXAMPLES:

sage: [sloane.A000142._eval(n) for n in range(10)]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
_repr_()

EXAMPLES:

sage: sloane.A000142._repr_()
'Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).'
class sage.combinat.sloane_functions.A000153
__init__()

a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.

With offset 1, permanent of (0,1)-matrix of size n \times (n+d) with d=2 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000153; a
a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
82508
sage: a(20)
10315043624498196944
sage: a.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]

AUTHORS:

  • Jaap Spies (2007-01-13)
_repr_()

EXAMPLES:

sage: sloane.A000153._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.'
class sage.combinat.sloane_functions.A000165
__init__()

Double factorial numbers: (2n)!! = 2^n*n!.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000165;a
Double factorial numbers: (2n)!! = 2^n*n!.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
10321920
sage: a(20)
2551082656125828464640000
sage: a.list(9)
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

AUTHORS:

  • Jaap Spies (2007-01-24)
_eval(n)

EXAMPLES:

sage: [sloane.A000165._eval(n) for n in range(10)]
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560]
_repr_()

EXAMPLES:

sage: sloane.A000165._repr_()
'Double factorial numbers: (2n)!! = 2^n*n!.'
class sage.combinat.sloane_functions.A000166
__init__()

Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

With offset 1 also the permanent of a (0,1)-matrix of order n with n 0’s not on a line.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000166;a
Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
sage: a(0)
1
sage: a(1)
0
sage: a(2)
1
sage: a.offset
0
sage: a(8)
14833
sage: a(20)
895014631192902121
sage: a.list(9)
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A000166._eval(n) for n in range(9)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]
_repr_()

EXAMPLES:

sage: sloane.A000166._repr_()
'Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.'
class sage.combinat.sloane_functions.A000169
__init__()

Number of labeled rooted trees with n nodes: n^{(n-1)}.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000169;a
Number of labeled rooted trees with n nodes: n^(n-1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(10)
1000000000
sage: a.list(11)
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000169._eval(n) for n in range(1,11)]
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000]
_repr_()

EXAMPLES:

sage: sloane.A000169._repr_()
'Number of labeled rooted trees with n nodes: n^(n-1).'
class sage.combinat.sloane_functions.A000203
__init__()

The sequence \sigma(n), where \sigma(n) is the sum of the divisors of n. Also called \sigma_1(n).

The function sigma(n, k) implements \sigma_k(n) in Sage.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000203; a
sigma(n) = sum of divisors of n. Also called sigma_1(n).
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(256)
511
sage: a.list(12)
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A000203._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
_repr_()

EXAMPLES:

sage: sloane.A000203._repr_()
'sigma(n) = sum of divisors of n. Also called sigma_1(n).'
class sage.combinat.sloane_functions.A000204
__init__()

Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000204; a
Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A000204._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
_repr_()

EXAMPLES:

sage: sloane.A000204._repr_()
'Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.'
class sage.combinat.sloane_functions.A000213
__init__()

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000213;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1
sage: a(11)
355
sage: a.list(12)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

AUTHORS:

  • Jaap Spies (2007-01-19)
_eval(n)

EXAMPLES:

sage: [sloane.A000213._eval(n) for n in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
_precompute(how_many=20)

EXAMPLES:

sage: initial = len(sloane.A000213._b)
sage: sloane.A000213._precompute(10)
sage: len(sloane.A000213._b) - initial == 10
True
_repr_()

EXAMPLES:

sage: sloane.A000213._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
list(n)

EXAMPLES:

sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
class sage.combinat.sloane_functions.A000217
__init__()

Triangular numbers: a(n) = {n+1} \choose 2) = n(n+1)/2.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000217;a
Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
sage: a(0)
0
sage: a(2)
3
sage: a(8)
36
sage: a(2000)
2001000
sage: a.list(9)
[0, 1, 3, 6, 10, 15, 21, 28, 36]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000217._eval(n) for n in range(10)]
[0, 1, 3, 6, 10, 15, 21, 28, 36, 45]
_repr_()

EXAMPLES:

sage: sloane.A000217._repr_()
'Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.'
class sage.combinat.sloane_functions.A000225
__init__()

2^n - 1.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000225;a
2^n - 1.
sage: a(0)
0
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(12)
4095
sage: a.list(12)
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000225._eval(n) for n in range(10)]
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511]
_repr_()

EXAMPLES:

sage: sloane.A000225._repr_()
'2^n - 1.'
class sage.combinat.sloane_functions.A000244
__init__()

Powers of 3: a(n) = 3^n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000244;a
Powers of 3: a(n) = 3^n.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(3)
27
sage: a(11)
177147
sage: a.list(12)
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000244._eval(n) for n in range(10)]
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
_repr_()

EXAMPLES:

sage: sloane.A000244._repr_()
'Powers of 3: a(n) = 3^n.'
class sage.combinat.sloane_functions.A000255
__init__()

a(n) = n*a(n-1) + (n-1)*a(n-2), with a(0) = 1, a(1) = 1.

With offset 1, permanent of (0,1)-matrix of size n \times (n+d) with d=1 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000255;a
a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
sage: a(0)
1
sage: a(1)
1
sage: a.offset
0
sage: a(8)
148329
sage: a(22)
9923922230666898717143
sage: a.list(9)
[1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

AUTHORS:

  • Jaap Spies (2007-01-13)
_repr_()

EXAMPLES:

sage: sloane.A000255._repr_()
'a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.'
class sage.combinat.sloane_functions.A000261
__init__()

a(n) = n*a(n-1) + (n-3)*a(n-2), with a(1) = 1, a(2) = 1.

With offset 1, permanent of (0,1)-matrix of size n \times (n+d) with d=3 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000261;a
a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a.offset
1
sage: a(8)
30637
sage: a(22)
1801366114380914335441
sage: a.list(9)
[0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A000261._repr_()
'a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.'
class sage.combinat.sloane_functions.A000272
__init__()

Number of labeled rooted trees on n nodes: n^{(n-2)}.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(11)
[1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000272._eval(n) for n in range(1,11)]
[1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000]
_repr_()

EXAMPLES:

sage: sloane.A000272._repr_()
'Number of labeled rooted trees with n nodes: n^(n-2).'
class sage.combinat.sloane_functions.A000290
__init__()

The squares: a(n) = n^2.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000290;a
The squares: a(n) = n^2.
sage: a(0)
0
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(16)
256
sage: a.list(17)
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]       

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000290._eval(n) for n in range(10)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
_repr_()

EXAMPLES:

sage: sloane.A000290._repr_()
'The squares: a(n) = n^2.'
class sage.combinat.sloane_functions.A000292
__init__()

Tetrahedral (or pyramidal) numbers: {n+2} \choose 3 = n(n+1)(n+2)/6.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000292;a
Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
sage: a(0)
0
sage: a(2)
4
sage: a(11)
286
sage: a.list(12)
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000292._eval(n) for n in range(10)]
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165]
_repr_()

EXAMPLES:

sage: sloane.A000292._repr_()
'Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.'
class sage.combinat.sloane_functions.A000302
__init__()

Powers of 4: a(n) = 4^n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000302;a
Powers of 4: a(n) = 4^n.
sage: a(0)
1
sage: a(1)
4
sage: a(2)
16
sage: a(10)
1048576
sage: a.list(12)
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000302._eval(n) for n in range(10)]
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]
_repr_()

EXAMPLES:

sage: sloane.A000302._repr_()
'Powers of 4: a(n) = 4^n.'
class sage.combinat.sloane_functions.A000312
__init__()

Number of labeled mappings from n points to themselves (endofunctions): n^n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000312;a
Number of labeled mappings from n points to themselves (endofunctions): n^n.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(1)
1
sage: a(9)
387420489
sage: a.list(11)
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]       

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000312._eval(n) for n in range(10)]
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489]
_repr_()

EXAMPLES:

sage: sloane.A000312._repr_()
'Number of labeled mappings from n points to themselves (endofunctions): n^n.'
class sage.combinat.sloane_functions.A000326
__init__()

Pentagonal numbers: n(3n-1)/2.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000326;a
Pentagonal numbers: n(3n-1)/2.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
5
sage: a(10)
145
sage: a.list(12)
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000326._eval(n) for n in range(10)]
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117]
_repr_()

EXAMPLES:

sage: sloane.A000326._repr_()
'Pentagonal numbers: n(3n-1)/2.'
class sage.combinat.sloane_functions.A000330
__init__()

Square pyramidal numbers” 0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000330;a
Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
14
sage: a(11)
506
sage: a.list(12)
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000330._eval(n) for n in range(10)]
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285]
_repr_()

EXAMPLES:

sage: sloane.A000330._repr_()
'Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.'
class sage.combinat.sloane_functions.A000396
__init__()

Perfect numbers: equal to sum of proper divisors.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000396;a
Perfect numbers: equal to sum of proper divisors.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
28
sage: a(7)
137438691328
sage: a.list(7)
[6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000396._eval(n) for n in range(1,6)]
[6, 28, 496, 8128, 33550336]
_repr_()

EXAMPLES:

sage: sloane.A000396._repr_()
'Perfect numbers: equal to sum of proper divisors.'
class sage.combinat.sloane_functions.A000578
__init__()

The cubes: a(n) = n^3.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000578;a
The cubes: n^3
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
27
sage: a(11)
1331
sage: a.list(12)
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000578._eval(n) for n in range(10)]
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729]
_repr_()

EXAMPLES:

sage: sloane.A000578._repr_()
'The cubes: n^3'
class sage.combinat.sloane_functions.A000583
__init__()

Fourth powers: a(n) = n^4.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000583;a
Fourth powers: n^4.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
16
sage: a(9)
6561
sage: a.list(10)
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

AUTHORS:

  • Jaap Spies (2007-02-04)
_eval(n)

EXAMPLES:

sage: [sloane.A000583._eval(n) for n in range(10)]
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]
_repr_()

EXAMPLES:

sage: sloane.A000583._repr_()
'Fourth powers: n^4.'
class sage.combinat.sloane_functions.A000587
__init__()

The sequence of Uppuluri-Carpenter numbers.

The Uppuluri-Carpenter number C_n counts the imbalance in the number of ways to put n distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty.

Let S(n, k) denote the Stirling number of the second kind. Then

C_n = \sum{k=0}^{n} (-1)^k S(n, k) .

INPUT:

  • n - integer = 0

OUTPUT:

  • integer - C_n

EXAMPLES:

sage: a = sloane.A000587; a
Sequence of Uppuluri-Carpenter numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
sage: a.list(10)
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

AUTHORS:

  • Nick Alexander
_repr_()

EXAMPLES:

sage: sloane.A000587._repr_()
'Sequence of Uppuluri-Carpenter numbers'
class sage.combinat.sloane_functions.A000668
__init__()

Mersenne primes (of form 2^p - 1 where p is a prime).

(See A000043 for the values of p.)

Warning: a(39) has 4,053,946 digits!

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000668;a
Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
sage: a(1)
3
sage: a(2)
7
sage: a(12)
170141183460469231731687303715884105727

Warning: a(39) has 4,053,946 digits!

sage: a(40)
...
IndexError: list index out of range
sage: a.list(8)
[3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000668._eval(n) for n in range(1,11)]
[3,
        7,
        31,
        127,
        8191,
        131071,
        524287,
        2147483647,
        2305843009213693951,
        618970019642690137449562111]
_repr_()

EXAMPLES:

sage: sloane.A000668._repr_()
'Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)'
class sage.combinat.sloane_functions.A000670
__init__()

Number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000670;a
Number of preferential arrangements of n labeled elements.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(9)
7087261
sage: a.list(10)
[1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

AUTHORS:

  • Jaap Spies (2007-02-03)
_eval(n)

EXAMPLES:

sage: [sloane.A000670._eval(n) for n in range(1,10)]
[1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
_repr_()

EXAMPLES:

sage: sloane.A000670._repr_()
'Number of preferential arrangements of n labeled elements.'
class sage.combinat.sloane_functions.A000720
__init__()

pi(n), the number of primes \le n. Sometimes called PrimePi(n).

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000720;a
pi(n), the number of primes <= n. Sometimes called PrimePi(n)
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a(8)
4
sage: a(1000)
168
sage: a.list(12)
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000720._eval(n) for n in range(1,11)]
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4]
_repr_()

EXAMPLES:

sage: sloane.A000720._repr_()
'pi(n), the number of primes <= n. Sometimes called PrimePi(n)'
class sage.combinat.sloane_functions.A000796
__init__()

Decimal expansion of \pi.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000796;a
Decimal expansion of Pi.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(13)
9
sage: a.list(14)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
sage: a(100)
7

AUTHOR:

  • Jaap Spies (2007-01-30)
_eval(n)

EXAMPLES:

sage: [sloane.A000796._eval(n) for n in range(1,11)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
_precompute(how_many=1000)

EXAMPLES:

sage: initial = len(sloane.A000796._b)
sage: sloane.A000796._precompute(10)
sage: len(sloane.A000796._b) - initial
10
_repr_()

EXAMPLES:

sage: sloane.A000796._repr_()
'Decimal expansion of Pi.'
list(n)

EXAMPLES:

sage: sloane.A000796.list(10)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
pi()

Based on an algorithm of Lambert Meertens The ABC-programming language!!!

EXAMPLES:

sage: it = sloane.A000796.pi()
sage: [it.next() for i in range(10)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
class sage.combinat.sloane_functions.A000961
__init__()

Prime powers

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000961;a
Prime powers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A000961._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A000961._b)
sage: sloane.A000961._precompute()
sage: len(sloane.A000961._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A000961._repr_()
'Prime powers.'
list(n)

EXAMPLES:

sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
class sage.combinat.sloane_functions.A000984
__init__()

Central binomial coefficients: 2n \choose n = \frac {(2n)!} {(n!)^2}.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000984;a
Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
sage: a(0)
1
sage: a(2)
6
sage: a(8)
12870
sage: a.list(9)
[1, 2, 6, 20, 70, 252, 924, 3432, 12870]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A000984._eval(n) for n in range(10)]
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
_repr_()

EXAMPLES:

sage: sloane.A000984._repr_()
'Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2'
class sage.combinat.sloane_functions.A001006
__init__()

Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001006;a
Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
15511
sage: a.list(13)
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]       

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A001006._eval(n) for n in range(10)]
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
_repr_()

EXAMPLES:

sage: sloane.A001006._repr_()
'Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.'
class sage.combinat.sloane_functions.A001045
__init__()

Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2), a(0) = 0 and a(1) = 1.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001045;a
Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(11)
683
sage: a.list(12)
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

AUTHORS:

  • Jaap Spies (2007-01-26)
_repr_()

EXAMPLES:

sage: sloane.A001045._repr_()
'Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).'
class sage.combinat.sloane_functions.A001055
__init__()

Number of ways of factoring n with all factors 1.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001055;a
Number of ways of factoring n with all factors >1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(9)
2
sage: a.list(16)
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

AUTHORS:

  • Jaap Spies (2007-02-04)
_eval(n)

EXAMPLES:

sage: [sloane.A001055._eval(n) for n in range(1,11)]
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2]
_repr_()

EXAMPLES:

sage: sloane.A001055._repr_()
'Number of ways of factoring n with all factors >1.'
nwf(n, m)

EXAMPLES:

sage: sloane.A001055.nwf(4,1)
0
sage: sloane.A001055.nwf(4,2)
1
sage: sloane.A001055.nwf(4,3)
1
sage: sloane.A001055.nwf(4,4)
2
class sage.combinat.sloane_functions.A001109
__init__()

a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001109;a
a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
235416
sage: a(60)
1515330104844857898115857393785728383101709300
sage: a.list(9)
[0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

AUTHORS:

  • Jaap Spies (2007-01-24)
_repr_()

EXAMPLES:

sage: sloane.A001109._repr_()
'a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1'
class sage.combinat.sloane_functions.A001110
__init__()

Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001110; a
Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
55420693056
sage: a(21)
4446390382511295358038307980025
sage: a.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A001110._repr_()
'Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.'
g(k)

EXAMPLES:

sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0
class sage.combinat.sloane_functions.A001147
__init__()

Double factorial numbers: (2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001147;a
Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
sage: a(0)
1
sage: a.offset
0
sage: a(8)
2027025
sage: a(20)
319830986772877770815625
sage: a.list(9)
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

AUTHORS:

  • Jaap Spies (2007-01-24)
_eval(n)

EXAMPLES:

sage: [sloane.A001147._eval(n) for n in range(10)]
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
_repr_()

EXAMPLES:

sage: sloane.A001147._repr_()
'Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).'
class sage.combinat.sloane_functions.A001157
__init__()

The sequence \sigma_2(n), sum of squares of divisors of n.

The function sigma(n, k) implements \sigma_k* in Sage.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001157;a
sigma_2(n): sum of squares of divisors of n
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
5
sage: a(8)
85
sage: a.list(9)
[1, 5, 10, 21, 26, 50, 50, 85, 91]

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A001157._eval(n) for n in range(1,11)]
[1, 5, 10, 21, 26, 50, 50, 85, 91, 130]
_repr_()

EXAMPLES:

sage: sloane.A001157._repr_()
'sigma_2(n): sum of squares of divisors of n'
class sage.combinat.sloane_functions.A001189
__init__()

Number of degree-n permutations of order exactly 2.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001189;a
Number of degree-n permutations of order exactly 2.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(2)
1
sage: a(12)
140151
sage: a.list(13)
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

AUTHORS:

  • Jaap Spies (2007-02-03)
_eval(n)

EXAMPLES:

sage: [sloane.A001189._eval(n) for n in range(1,11)]
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495]
_repr_()

EXAMPLES:

sage: sloane.A001189._repr_()
'Number of degree-n permutations of order exactly 2.'
class sage.combinat.sloane_functions.A001221
__init__()

Number of different prime divisors of n

Also called omega(n) or \omega(n). Maximal number of terms in any factorization of n. Number of prime powers that divide n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001221; a
Number of distinct primes dividing n (also called omega(n)).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
1
sage: a(41)
1
sage: a(84792)
3
sage: a.list(12)
[0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

AUTHORS:

  • Jaap Spies (2007-01-19)
_eval(n)

EXAMPLES:

sage: [sloane.A001221._eval(n) for n in range(1,10)]
[0, 1, 1, 1, 1, 2, 1, 1, 1]
_repr_()

EXAMPLES:

sage: sloane.A001221._repr_()
'Number of distinct primes dividing n (also called omega(n)).'
class sage.combinat.sloane_functions.A001222
__init__()

Number of prime divisors of n (counted with multiplicity).

Also called bigomega(n) or \Omega(n). Maximal number of terms in any factorization of n. Number of prime powers that divide n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001222; a
Number of prime divisors of n (counted with multiplicity).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
3
sage: a(41)
1
sage: a(84792)
5
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

AUTHORS:

  • Jaap Spies (2007-01-19)
_eval(n)

EXAMPLES:

sage: [sloane.A001222._eval(n) for n in range(1,10)]
[0, 1, 1, 2, 1, 2, 1, 3, 2]
_repr_()

EXAMPLES:

sage: sloane.A001222._repr_()
'Number of prime divisors of n (counted with multiplicity).'
class sage.combinat.sloane_functions.A001227
__init__()

Number of odd divisors of n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001227; a
Number of odd divisors of n
sage: a.offset
1
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
3
sage: a(256)
1
sage: a(29)
2
sage: a.list(20)
[1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
_eval(n)

EXAMPLES:

sage: [sloane.A001227._eval(n) for n in range(1,10)]
[1, 1, 2, 1, 2, 2, 2, 1, 3]
_repr_()

EXAMPLES:

sage: sloane.A001227._repr_()
'Number of odd divisors of n'
class sage.combinat.sloane_functions.A001333
__init__()

Numerators of continued fraction convergents to \sqrt 2.

See also A000129

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001333;a
Numerators of continued fraction convergents to sqrt(2).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(3)
7
sage: a(11)
8119
sage: a.list(12)
[1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

AUTHORS:

  • Jaap Spies (2007-02-01)
_repr_()

EXAMPLES:

sage: sloane.A001333._repr_()
'Numerators of continued fraction convergents to sqrt(2).'
class sage.combinat.sloane_functions.A001358
__init__()

Products of two primes.

These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001358;a
Products of two primes.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(8)
22
sage: a(200)
669
sage: a.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A001358._eval(n) for n in range(1,10)]
[4, 6, 9, 10, 14, 15, 21, 22, 25]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A001358._b)
sage: sloane.A001358._precompute()
sage: len(sloane.A001358._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A001358._repr_()
'Products of two primes.'
list(n)

EXAMPLES:

sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]
class sage.combinat.sloane_functions.A001405
__init__()

Central binomial coefficients: n \choose \lfloor \frac {n}{ 2} \rfloor.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001405;a
Central binomial coefficients: C(n,floor(n/2)).
sage: a(0)
1
sage: a(2)
2
sage: a(12)
924
sage: a.list(12)
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A001405._eval(n) for n in range(10)]
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
_repr_()

EXAMPLES:

sage: sloane.A001405._repr_()
'Central binomial coefficients: C(n,floor(n/2)).'
class sage.combinat.sloane_functions.A001477
__init__()

The nonnegative integers.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001477;a
The nonnegative integers.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3382789)
3382789
sage: a(11)
11
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]       

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A001477._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
_repr_()

EXAMPLES:

sage: sloane.A001477._repr_()
'The nonnegative integers.'
class sage.combinat.sloane_functions.A001694
__init__()

This function returns the n-th Powerful Number:

A positive integer n is powerful if for every prime p dividing n, p^2 also divides n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001694; a
Powerful Numbers (also called squarefull, square-full or 2-full numbers).
sage: a.offset
1
sage: a(1)
1
sage: a(4)
9
sage: a(100)
3136
sage: a(156)
7225
sage: a.list(19)
[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
_eval(n)

EXAMPLES:

sage: [sloane.A001694._eval(n) for n in range(1,10)]
[1, 4, 8, 9, 16, 25, 27, 32, 36]
_powerful_numbers_in_range(n, m)

EXAMPLES:

sage: sloane.A001694._powerful_numbers_in_range(0,50)
[4, 8, 9, 16, 25, 27, 32, 36, 49]
_precompute(how_many=10000)

EXAMPLES:

sage: initial = len(sloane.A001694._b)
sage: sloane.A001694._precompute()
sage: len(sloane.A001694._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A001694._repr_()
'Powerful Numbers (also called squarefull, square-full or 2-full numbers).'
is_powerful(n)

This function returns True if and only if n is a Powerful Number:

A positive integer n is powerful if for every prime p dividing n, p^2 also divides n. See Sloane’s OEIS A001694.

INPUT:

  • n - integer

OUTPUT:

  • True - if n is a Powerful number, else False

EXAMPLES:

sage: a = sloane.A001694
sage: a.is_powerful(2500)
True
sage: a.is_powerful(20)
False

AUTHORS:

  • Jaap Spies (2006-12-07)
list(n)

EXAMPLES:

sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]
class sage.combinat.sloane_functions.A001836
__init__()

Numbers n such that \phi(2n-1) < \phi(2n), where \phi is Euler’s totient function.

Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001836; a
Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
sage: a.offset
1
sage: a(1)
53
sage: a(8)
683
sage: a(300)
17798
sage: a.list(12)
[53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer

Compare: Searching Sloane’s online database... Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

AUTHORS:

  • Jaap Spies (2007-01-17)
_eval(n)

EXAMPLES:

sage: [sloane.A001836._eval(n) for n in range(1,10)]
[53, 83, 158, 263, 293, 368, 578, 683, 743]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A001836._b)
sage: sloane.A001836._precompute()
sage: len(sloane.A001836._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A001836._repr_()
"Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010."
list(n)

EXAMPLES:

sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]
class sage.combinat.sloane_functions.A001906
__init__()

F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001906; a
F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
sage: a(0) 
0       
sage: a(1)
1
sage: a(8)
987
sage: a(22)
701408733
sage: a.list(12)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A001906._repr_()
'F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).'
class sage.combinat.sloane_functions.A001909
__init__()

a(n) = n*a(n-1) + (n-4)*a(n-2), with a(2) = 0, a(3) = 1.

With offset 1, permanent of (0,1)-matrix of size n \times (n+d) with d=4 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer = 2

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001909;a
a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
sage: a(1)
...
ValueError: input n (=1) must be an integer >= 2
sage: a.offset
2
sage: a(2)
0
sage: a(8)
8544
sage: a(22)
470033715095287415734
sage: a.list(9)
[0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

AUTHORS:

  • Jaap Spies (2007-01-13)
_repr_()

EXAMPLES:

sage: sloane.A001909._repr_()
'a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.'
class sage.combinat.sloane_functions.A001910
__init__()

a(n) = n*a(n-1) + (n-5)*a(n-2), with a(3) = 0, a(4) = 1.

With offset 1, permanent of (0,1)-matrix of size n \times (n+d) with d=5 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer = 3

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001910;a
a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
sage: a(0)
...
ValueError: input n (=0) must be an integer >= 3
sage: a(3)
0
sage: a.offset
3
sage: a(8)
1909
sage: a(22)
98125321641110663023
sage: a.list(9)
[0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]       

AUTHORS:

  • Jaap Spies (2007-01-13)
_repr_()

EXAMPLES:

sage: sloane.A001910._repr_()
'a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.'
class sage.combinat.sloane_functions.A001969
__init__()

Evil numbers: even number of 1’s in binary expansion.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001969;a
Evil numbers: even number of 1's in binary expansion.
sage: a(0)
0
sage: a(1)
3
sage: a(2)
5
sage: a(12)
24
sage: a.list(13)
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A001969._eval(n) for n in range(10)]
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18]
_repr_()

EXAMPLES:

sage: sloane.A001969._repr_()
"Evil numbers: even number of 1's in binary expansion."
class sage.combinat.sloane_functions.A002110
__init__()

Primorial numbers (first definition): product of first n primes. Sometimes written p\#.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002110;a
Primorial numbers (first definition): product of first n primes. Sometimes written p#.
sage: a(0)
1
sage: a(2)
6
sage: a(8)
9699690
sage: a(17)
1922760350154212639070
sage: a.list(9)
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A002110._eval(n) for n in range(10)]
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870]
_repr_()

EXAMPLES:

sage: sloane.A002110._repr_()
'Primorial numbers (first definition): product of first n primes. Sometimes written p#.'
class sage.combinat.sloane_functions.A002113
__init__()

Palindromes in base 10.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002113;a
Palindromes in base 10.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(12)
33
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A002113._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A002113._b)
sage: sloane.A002113._precompute()
sage: len(sloane.A002113._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A002113._repr_()
'Palindromes in base 10.'
list(n)

EXAMPLES:

sage: sloane.A002113.list(15)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
class sage.combinat.sloane_functions.A002275
__init__()

Repunits: \frac {(10^n - 1)}{9}. Often denoted by R_n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002275;a
Repunits: (10^n - 1)/9. Often denoted by R_n.
sage: a(0)
0
sage: a(2)
11
sage: a(8)
11111111
sage: a(20)
11111111111111111111
sage: a.list(9)
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A002275._eval(n) for n in range(10)]
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111]
_repr_()

EXAMPLES:

sage: sloane.A002275._repr_()
'Repunits: (10^n - 1)/9. Often denoted by R_n.'
class sage.combinat.sloane_functions.A002378
__init__()

Oblong (or pronic, or heteromecic) numbers: n(n+1).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002378;a
Oblong (or pronic, or heteromecic) numbers: n(n+1).
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(1)
2
sage: a(11)
132
sage: a.list(12)
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A002378._eval(n) for n in range(10)]
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90]
_repr_()

EXAMPLES:

sage: sloane.A002378._repr_()
'Oblong (or pronic, or heteromecic) numbers: n(n+1).'
class sage.combinat.sloane_functions.A002620
__init__()

Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \lfloor n^2/4 \rfloor.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002620;a
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
25
sage: a.list(12)
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A002620._eval(n) for n in range(10)]
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20]
_repr_()

EXAMPLES:

sage: sloane.A002620._repr_()
'Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).'
class sage.combinat.sloane_functions.A002808
__init__()

The composite numbers: numbers n of the form xy for x > 1 and y > 1.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A002808;a
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(11)
20
sage: a.list(12)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A002808._eval(n) for n in range(1,11)]
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A002808._b)
sage: sloane.A002808._precompute()
sage: len(sloane.A002808._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A002808._repr_()
'The composite numbers: numbers n of the form x*y for x > 1 and y > 1.'
list(n)

EXAMPLES:

sage: sloane.A002808.list(10)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
class sage.combinat.sloane_functions.A003418
__init__()

Least common multiple (or lcm) of \{1, 2, \cdots, n\}.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A003418;a
Least common multiple (or lcm) of {1, 2, ..., n}.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
360360
sage: a.list(14)
[1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
sage: a(20.0)
...
TypeError: input must be an int, long, or Integer

AUTHOR:

  • Jaap Spies (2007-01-31)
_eval(n)

EXAMPLES:

sage: [sloane.A003418._eval(n) for n in range(1,11)]
[1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520]
_repr_()

EXAMPLES:

sage: sloane.A003418._repr_()
'Least common multiple (or lcm) of {1, 2, ..., n}.'
class sage.combinat.sloane_functions.A004086
__init__()

Read n backwards (referred to as R(n) in many sequences).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A004086;a
Read n backwards (referred to as R(n) in many sequences).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(3333)
3333
sage: a(12345)
54321
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A004086._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
_repr_()

EXAMPLES:

sage: sloane.A004086._repr_()
'Read n backwards (referred to as R(n) in many sequences).'
class sage.combinat.sloane_functions.A004526
__init__()

The nonnegative integers repeated

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A004526;a
The nonnegative integers repeated.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
5
sage: a.list(12)
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A004526._eval(n) for n in range(10)]
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4]
_repr_()

EXAMPLES:

sage: sloane.A004526._repr_()
'The nonnegative integers repeated.'
class sage.combinat.sloane_functions.A005100
__init__()

Deficient numbers: \sigma(n) < 2n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005100;a
Deficient numbers: sigma(n) < 2n
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(12)
14
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A005100._eval(n) for n in range(1,10)]
[1, 2, 3, 4, 5, 7, 8, 9, 10]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A005100._b)
sage: sloane.A005100._precompute()
sage: len(sloane.A005100._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A005100._repr_()
'Deficient numbers: sigma(n) < 2n'
list(n)

EXAMPLES:

sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
class sage.combinat.sloane_functions.A005101
__init__()

Abundant numbers (sum of divisors of n exceeds 2n).

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005101;a
Abundant numbers (sum of divisors of n exceeds 2n).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
12
sage: a(2)
18
sage: a(12)
60
sage: a.list(12)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A005101._eval(n) for n in range(1,11)]
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A005101._b)
sage: sloane.A005101._precompute()
sage: len(sloane.A005101._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A005101._repr_()
'Abundant numbers (sum of divisors of n exceeds 2n).'
list(n)

EXAMPLES:

sage: sloane.A005101.list(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
class sage.combinat.sloane_functions.A005117
__init__()

Square-free numbers

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005117;a
Square-free numbers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A005117._eval(n) for n in range(1,11)]
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A005117._b)
sage: sloane.A005117._precompute()
sage: len(sloane.A005117._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A005117._repr_()
'Square-free numbers.'
list(n)

EXAMPLES:

sage: sloane.A005117.list(10)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
class sage.combinat.sloane_functions.A005408
__init__()

The odd numbers a(n) = 2n + 1.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005408;a
The odd numbers a(n) = 2n + 1.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(4)
9
sage: a(11)
23
sage: a.list(12)
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

AUTHORS:

  • Jaap Spies (2007-01-26)
_eval(n)

EXAMPLES:

sage: [sloane.A005408._eval(n) for n in range(10)]
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19]
_repr_()

EXAMPLES:

sage: sloane.A005408._repr_()
'The odd numbers a(n) = 2n + 1.'
class sage.combinat.sloane_functions.A005843
__init__()

The even numbers: a(n) = 2n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005843;a
The even numbers: a(n) = 2n.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
2
sage: a(2)
4
sage: a(9)
18
sage: a.list(10)
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

AUTHORS:

  • Jaap Spies (2007-02-03)
_eval(n)

EXAMPLES:

sage: [sloane.A005843._eval(n) for n in range(10)]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
_repr_()

EXAMPLES:

sage: sloane.A005843._repr_()
'The even numbers: a(n) = 2n.'
class sage.combinat.sloane_functions.A006318
__init__()

Large Schroeder numbers.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A006318;a
Large Schroeder numbers.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
6
sage: a(9)
206098
sage: a.list(10)
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

AUTHORS:

  • Jaap Spies (2007-02-03)
_eval(n)

EXAMPLES:

sage: [sloane.A006318._eval(n) for n in range(10)]
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]
_repr_()

EXAMPLES:

sage: sloane.A006318._repr_()
'Large Schroeder numbers.'
class sage.combinat.sloane_functions.A006530
__init__()

Largest prime dividing n (with a(1)=1).

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A006530;a
Largest prime dividing n (with a(1)=1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(8)
2
sage: a(11)
11
sage: a.list(15)
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A006530._eval(n) for n in range(1,11)]
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5]
_repr_()

EXAMPLES:

sage: sloane.A006530._repr_()
'Largest prime dividing n (with a(1)=1).'
class sage.combinat.sloane_functions.A006882
__init__()

Double factorials n!!: a(n)=n \cdot a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A006882;a
Double factorials n!!: a(n)=n*a(n-2).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
384
sage: a(20)
3715891200
sage: a.list(9)
[1, 1, 2, 3, 8, 15, 48, 105, 384]

AUTHORS:

  • Jaap Spies (2007-01-24)
_eval(n)

EXAMPLES:

sage: [sloane.A006882._eval(n) for n in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
_precompute(how_many=10)

EXAMPLES:

sage: initial = len(sloane.A006882._b)
sage: sloane.A006882._precompute(10)
sage: len(sloane.A006882._b) - initial == 10
True
_repr_()

EXAMPLES:

sage: sloane.A006882._repr_()
'Double factorials n!!: a(n)=n*a(n-2).'
df()

Double factorials n!!: a(n)=n*a(n-2).

EXAMPLES:

sage: it = sloane.A006882.df()
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
list(n)

EXAMPLES:

sage: sloane.A006882.list(10)
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
class sage.combinat.sloane_functions.A007318
__init__()

Pascal’s triangle read by rows: C(n,k) = {n \choose k} = \frac {n!} {(k!(n-k)!)}, 0 \le k \le n.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A007318
sage: a(0)
1
sage: a(1)
1
sage: a(13)
4
sage: a.list(15)
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
sage: a(100)
715

AUTHORS:

  • Jaap Spies (2007-01-31)
_eval(n)

EXAMPLES:

sage: [sloane.A007318._eval(n) for n in range(10)]
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
_repr_()

EXAMPLES:

sage: sloane.A007318._repr_()
"Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n."
class sage.combinat.sloane_functions.A008275
__init__()

Triangle of Stirling numbers of first kind, s(n,k), n \ge 1, 1 \le k \le n.

The unsigned numbers are also called Stirling cycle numbers:

|s(n,k)| = number of permutations of n objects with exactly k cycles.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A008275;a
Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
-1
sage: a(3)
1
sage: a(11)
24
sage: a.list(12)
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A008275._eval(n) for n in range(1, 11)]
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1]
_repr_()

EXAMPLES:

sage: sloane.A008275._repr_()
'Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.'
s(n, k)

EXAMPLES:

sage: sloane.A008275.s(4,2)
11
sage: sloane.A008275.s(5,2)
-50
sage: sloane.A008275.s(5,3)
35
class sage.combinat.sloane_functions.A008277
__init__()

Triangle of Stirling numbers of 2nd kind, S2(n,k), n \ge 1, 1 \le k \le n.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A008277;a
Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(3)
1
sage: a(4.0)
...
TypeError: input must be an int, long, or Integer
sage: a.list(15)
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

AUTHORS:

  • Jaap Spies (2007-01-31)
_eval(n)

EXAMPLES:

sage: [sloane.A008277._eval(n) for n in range(1,11)]
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1]
_repr_()

EXAMPLES:

sage: sloane.A008277._repr_()
'Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.'
s2(n, k)

Returns the Stirling number S2(n,k) of the 2nd kind.

EXAMPLES:

sage: sloane.A008277.s2(4,2)
7
class sage.combinat.sloane_functions.A008683
__init__()

Moebius function \mu(n).

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A008683;a
Moebius function mu(n).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
-1
sage: a(12)
0
sage: a.list(12)
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A008683._eval(n) for n in range(1,11)]
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1]
_repr_()

EXAMPLES:

sage: sloane.A008683._repr_()
'Moebius function mu(n).'
class sage.combinat.sloane_functions.A010060
__init__()

Thue-Morse sequence.

Let A_k denote the first 2^k terms; then A_0 = 0, and for k \ge 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0’s and 1’s.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A010060;a
Thue-Morse sequence.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(12)
0
sage: a.list(13)
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]       

AUTHORS:

  • Jaap Spies (2007-02-02)
_eval(n)

EXAMPLES:

sage: [sloane.A010060._eval(n) for n in range(10)]
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0]
_repr_()

EXAMPLES:

sage: sloane.A010060._repr_()
'Thue-Morse sequence.'
class sage.combinat.sloane_functions.A015521
__init__()

Linear 2nd order recurrence, a(0)=0, a(1)=1 and a(n) = 3 a(n-1) + 4 a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A015521; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
13107
sage: a(41)
967140655691703339764941
sage: a.list(12)
[0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A015521._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).'
class sage.combinat.sloane_functions.A015523
__init__()

Linear 2nd order recurrence, a(0)=0, a(1)=1 and a(n) = 3 a(n-1) + 5 a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A015523; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
17727
sage: a(41)
6173719566474529739091481
sage: a.list(12)
[0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A015523._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).'
class sage.combinat.sloane_functions.A015530
__init__()

Linear 2nd order recurrence, a(0)=0, a(1)=1 and a(n) = 4 a(n-1) + 3 a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A015530;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
41008
sage: a.list(9)
[0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A015530._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).'
class sage.combinat.sloane_functions.A015531
__init__()

Linear 2nd order recurrence, a(0)=0, a(1)=1 and a(n) = 4 a(n-1) + 5 a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A015531;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
65104
sage: a(60)
144560289664733924534327040115992228190104
sage: a.list(9)
[0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A015531._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).'
class sage.combinat.sloane_functions.A015551
__init__()

Linear 2nd order recurrence, a(0)=0, a(1)=1 and a(n) = 6 a(n-1) + 5 a(n-2).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A015551;a
Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
570216
sage: a(60)
7110606606530059736761484557155863822531970573036
sage: a.list(9)
[0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A015551._repr_()
'Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).'
class sage.combinat.sloane_functions.A018252
__init__()

The nonprime numbers, starting with 1.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A018252;a
The nonprime numbers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
4
sage: a(9)
15
sage: a.list(10)
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

AUTHORS:

  • Jaap Spies (2007-02-04)
_eval(n)

EXAMPLES:

sage: [sloane.A018252._eval(n) for n in range(1,11)]
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]
_repr_()

EXAMPLES:

sage: sloane.A018252._repr_()
'The nonprime numbers.'
class sage.combinat.sloane_functions.A020639
__init__()

Least prime dividing n with a(1)=1.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A020639;a
Least prime dividing n (a(1)=1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(13)
13
sage: a.list(14)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

AUTHORS:

  • Jaap Spies (2007-01-25)
_eval(n)

EXAMPLES:

sage: [sloane.A020639._eval(n) for n in range(1,11)]
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
_precompute(how_many=50)

EXAMPLES:

sage: initial = len(sloane.A020639._b)
sage: sloane.A020639._precompute(10)
sage: len(sloane.A020639._b) - initial == 10
True
_repr_()

EXAMPLES:

sage: sloane.A020639._repr_()
'Least prime dividing n (a(1)=1).'
list(n)

EXAMPLES:

sage: sloane.A020639.list(10)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
class sage.combinat.sloane_functions.A046660(offset=1)

Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

\Omega(n) - \omega(n).

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
2
sage: a(41)
0
sage: a(84792)
2
sage: a.list(12)
[0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

AUTHORS:

  • Jaap Spies (2007-01-19)
_eval(n)

EXAMPLES:

sage: [sloane.A046660._eval(n) for n in range(1,10)]
[0, 0, 0, 1, 0, 0, 0, 2, 1]
_repr_()

EXAMPLES:

sage: sloane.A046660._repr_()
'Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).'
class sage.combinat.sloane_functions.A049310
__init__()

Triangle of coefficients of Chebyshev’s S(n,x): U(n, \frac x 2) polynomials (exponents in increasing order).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A049310;a
Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
sage: a(0)
1
sage: a(1)
0
sage: a(13)
0
sage: a.list(15)
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
sage: a(200)
0
sage: a.keyword
['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

AUTHORS:

  • Jaap Spies (2007-01-31)
_eval(n)

EXAMPLES:

sage: [sloane.A049310._eval(n) for n in range(10)]
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1]
_repr_()

EXAMPLES:

sage: sloane.A049310._repr_()
"Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order)."
class sage.combinat.sloane_functions.A051959
__init__()

Linear second order recurrence. A051959.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A051959; a
Linear second order recurrence. A051959.
sage: a(0)
1
sage: a(1)
10
sage: a(8)
9969
sage: a(41)
42834431872413650
sage: a.list(12)
[1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A051959._repr_()
'Linear second order recurrence. A051959.'
g(k)

EXAMPLES:

sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0
class sage.combinat.sloane_functions.A055790
__init__()

a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A055790;a
a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
sage: a(0)
0
sage: a(1)
2
sage: a(2)
4
sage: a.offset
0
sage: a(8)
165016
sage: a(22)
10356214297533070441564
sage: a.list(9)
[0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A055790._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].'
class sage.combinat.sloane_functions.A061084
__init__()

Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A061084; a
Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
sage: a(0)
1
sage: a(1)
2
sage: a(8)
-29
sage: a(22)
-24476
sage: a.list(12)
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
sage: a.keyword
['sign', 'easy', 'nice']

AUTHORS:

  • Jaap Spies (2007-01-18)
_eval(n)

EXAMPLES:

sage: [sloane.A061084._eval(n) for n in range(10)]
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47]
_repr_()

EXAMPLES:

sage: sloane.A061084._repr_()
'Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).'
class sage.combinat.sloane_functions.A064553
__init__()

a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u \cdot v) = a(u) \cdot a(v) for u, v > 0.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A064553;a
a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(9)
9
sage: a.list(16)
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

AUTHORS:

  • Jaap Spies (2007-02-04)
_eval(n)

EXAMPLES:

sage: [sloane.A064553._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8]
_repr_()

EXAMPLES:

sage: sloane.A064553._repr_()
'a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0'
class sage.combinat.sloane_functions.A079922(offset=1)

function returns solutions to the Dancing School problem with n girls and n+3 boys.

The value is per(B), the permanent of the (0,1)-matrix B of size n \times n+3 with b(i,j)=1 if and only if i \le j \le i+n.

REFERENCES:

  • Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A079922; a
Solutions to the Dancing School problem with n girls and n+3 boys
sage: a.offset
1
sage: a(1)
4
sage: a(8)
2227
sage: a.list(8)
[4, 13, 36, 90, 212, 478, 1044, 2227]

Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]

sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
_eval(n)

EXAMPLES:

sage: [sloane.A079922._eval(n) for n in range(1,5)]
[4, 13, 36, 90]
_repr_()

EXAMPLES:

sage: sloane.A079922._repr_()
'Solutions to the Dancing School problem with n girls and n+3 boys'
class sage.combinat.sloane_functions.A079923(offset=1)

function returns solutions to the Dancing School problem with n girls and n+4 boys.

The value is per(B), the permanent of the (0,1)-matrix B of size n \times n+3 with b(i,j)=1 if and only if i \le j \le i+n.

REFERENCES:

  • Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A079923; a
Solutions to the Dancing School problem with n girls and n+4 boys
sage: a.offset
1
sage: a(1)
5
sage: a(8)
15458
sage: a.list(8)
[5, 21, 76, 246, 738, 2108, 5794, 15458]

Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]

sage: a(0)
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-17)
_eval(n)

EXAMPLES:

sage: [sloane.A079923._eval(n) for n in range(1,11)]
[5, 21, 76, 246, 738, 2108, 5794, 15458, 40296, 103129]
_repr_()

EXAMPLES:

sage: sloane.A079923._repr_()
'Solutions to the Dancing School problem with n girls and n+4 boys'
class sage.combinat.sloane_functions.A082411
__init__()

Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(0) = 407389224418, a(1) = 76343678551. This is the second-order linear recurrence sequence with a(0) and a(1) co-prime, that R. L. Graham in 1964 stated did not contain any primes.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A082411;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
76343678551
sage: a(2)
483732902969
sage: a(3)
560076581520
sage: a(20)
2219759332689173
sage: a.list(4)
[407389224418, 76343678551, 483732902969, 560076581520]        

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A082411._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A083103
__init__()

Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(0) = 1786772701928802632268715130455793, a(1) = 1059683225053915111058165141686995. This is the second-order linear recurrence sequence with a(0) and a(1) co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A083103;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
1059683225053915111058165141686995
sage: a(2)
2846455926982717743326880272142788
sage: a(3)
3906139152036632854385045413829783
sage: a.offset
0
sage: a(8)
45481392851206651551714764671352204
sage: a(20)
14639253684254059531823985143948191708
sage: a.list(4)
[1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A083103._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A083104
__init__()

Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(0) = 331635635998274737472200656430763, a(1) = 1510028911088401971189590305498785. This is the second-order linear recurrence sequence with a(0) and a(1) co-prime. It was found by Ronald Graham in 1990.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A083104;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(3)
3351693458175078679851381267428333
sage: a.offset
0
sage: a(8)
36021870400834012982120004949074404
sage: a(20)
11601914177621826012468849361236300628

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A083104._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A083105
__init__()

Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(0) = 62638280004239857, a(1) = 49463435743205655. This is the second-order linear recurrence sequence with a(0) and a(1) co-prime. It was found by Donald Knuth in 1990.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A083105;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
49463435743205655
sage: a(2)
112101715747445512
sage: a(3)
161565151490651167
sage: a.offset
0
sage: a(8)
1853029790662436896
sage: a(20)
596510791500513098192
sage: a.list(4)
[62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A083105._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A083216
__init__()

Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(0) = 20615674205555510, a(1) = 3794765361567513. This is a second-order linear recurrence sequence with a(0) and a(1) co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A083216; a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(0)
20615674205555510
sage: a(1)
3794765361567513
sage: a(8)
347693837265139403
sage: a(41)
2738025383211084205003383
sage: a.list(4)
[20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

AUTHORS:

  • Jaap Spies (2007-01-19)
_repr_()

EXAMPLES:

sage: sloane.A083216._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
class sage.combinat.sloane_functions.A090010
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=6 and n zeros not on a line.

` a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43`.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090010;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
43
sage: a.offset
1
sage: a(8)
67741129
sage: a(22)
192416593029158989003270143
sage: a.list(9)
[6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

AUTHORS:

  • Jaap Spies (2007-01-23)
_repr_()

EXAMPLES:

sage: sloane.A090010._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.'
class sage.combinat.sloane_functions.A090012
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=2 and n-1 zeros not on a line.

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2), a(1)=3 and a(2)=9

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090012;a
Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(2)
9
sage: a.offset
1
sage: a(8)
890901
sage: a(22)
129020386652297208795129
sage: a.list(9)
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

AUTHORS:

  • Jaap Spies (2007-01-23)
_eval(n)

EXAMPLES:

sage: [sloane.A090012._eval(n) for n in range(1,11)]
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465]
_repr_()

EXAMPLES:

sage: sloane.A090012._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.'
class sage.combinat.sloane_functions.A090013
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=3 and n-1 zeros not on a line.

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090013;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
4
sage: a(2)
16
sage: a.offset
1
sage: a(8)
3481096
sage: a(22)
1112998577171142607670336
sage: a.list(9)
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

AUTHORS:

  • Jaap Spies (2007-01-23)
_eval(n)

EXAMPLES:

sage: [sloane.A090013._eval(n) for n in range(1,11)]
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284, 514872176]
_repr_()

EXAMPLES:

sage: sloane.A090013._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
class sage.combinat.sloane_functions.A090014
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=4 and n-1 zeros not on a line.

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090014;a
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
5
sage: a(2)
25
sage: a.offset
1
sage: a(8)
11016595
sage: a(22)
7469733600354446865509725
sage: a.list(9)
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

AUTHORS:

  • Jaap Spies (2007-01-23)
_eval(n)

EXAMPLES:

sage: [sloane.A090014._eval(n) for n in range(1,11)]
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365, 1893369505]
_repr_()

EXAMPLES:

sage: sloane.A090014._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.'
class sage.combinat.sloane_functions.A090015
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=5 and n-1 zeros not on a line.

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090015;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
36
sage: a.offset
1
sage: a(8)
29976192
sage: a(22)
41552258517692116794936876
sage: a.list(9)
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

AUTHORS:

  • Jaap Spies (2007-01-23)
_eval(n)

EXAMPLES:

sage: [sloane.A090015._eval(n) for n in range(1,10)]
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
_repr_()

EXAMPLES:

sage: sloane.A090015._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
class sage.combinat.sloane_functions.A090016
__init__()

Permanent of (0,1)-matrix of size n \times (n+d) with d=6 and n-1 zeros not on a line.

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]

A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

  • Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A090016;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
7
sage: a(2)
49
sage: a.offset
1
sage: a(8)
72737161
sage: a(22)
199341969448774341802426289
sage: a.list(9)
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

AUTHORS:

  • Jaap Spies (2007-01-23)
_eval(n)

EXAMPLES:

sage: [sloane.A090016._eval(n) for n in range(1,10)]
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]
_repr_()

EXAMPLES:

sage: sloane.A090016._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.'
class sage.combinat.sloane_functions.A111774
__init__()

Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers.

Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n = n for k = 1).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A111774; a
Numbers that can be written as a sum of at least three consecutive positive integers.
sage: a(1)
6
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
141
sage: a(156)
209
sage: a(302)
386
sage: a.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A111774._eval(n) for n in range(1,11)]
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A111774._b)
sage: sloane.A111774._precompute()
sage: len(sloane.A111774._b) - initial > 0
True
_repr_()

EXAMPLES:

sage: sloane.A111774._repr_()
'Numbers that can be written as a sum of at least three consecutive positive integers.'
is_number_of_the_third_kind(n)

This function returns True if and only if n is a number of the third kind.

A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n = n for k = 1).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

  • n - positive integer

OUTPUT:

  • True - if n is not prime and not a power of 2 False -

EXAMPLES:

sage: a = sloane.A111774
sage: a.is_number_of_the_third_kind(6)
True
sage: a.is_number_of_the_third_kind(100)
True
sage: a.is_number_of_the_third_kind(16)
False
sage: a.is_number_of_the_third_kind(97)
False

AUTHORS:

  • Jaap Spies (2006-12-09)
list(n)

EXAMPLES:

sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
class sage.combinat.sloane_functions.A111775
__init__()

Number of ways n can be written as a sum of at least three consecutive integers.

Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. a(n) strongly depends on the number of odd divisors of n (A001227): Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Only one of the factors is odd. For each odd divisor of n there is a unique corresponding k, k=1 and k=2 must be excluded.

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A111775; a
Number of ways n can be written as a sum of at least three consecutive integers.
sage: a(1)
0
sage: a(0)
0

We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

sage: a(15)
2
sage: a(100)
2
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2006-12-09)
_eval(n)

EXAMPLES:

sage: [sloane.A111775._eval(n) for n in range(10)]
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
_repr_()

EXAMPLES:

sage: sloane.A111775._repr_()
'Number of ways n can be written as a sum of at least three consecutive integers.'
class sage.combinat.sloane_functions.A111776
__init__()

The n`th term of the sequence `a(n) is the largest k such that n can be written as sum of k consecutive integers.

n is the sum of at most a(n) consecutive positive integers. Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Only one of the factors is odd. For each odd divisor d of n there is a unique corresponding k = min(d,2n/d). a(n) is the largest among those k . See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A111776; a
a(n) is the largest k such that n can be written as sum of k consecutive integers.
sage: a(0)
1
sage: a(2)
1
sage: a.list(9)
[1, 1, 1, 2, 1, 2, 3, 2, 1]

AUTHORS:

  • Jaap Spies (2007-01-13)
_eval(n)

EXAMPLES:

sage: [sloane.A111776._eval(n) for n in range(10)]
[1, 1, 1, 2, 1, 2, 3, 2, 1, 3]
_repr_()

EXAMPLES:

sage: sloane.A111776._repr_()
'a(n) is the largest k such that n can be written as sum of k consecutive integers.'
class sage.combinat.sloane_functions.A111787
__init__()

This function returns the n-th number of Sloane’s sequence A111787

a(n)=0 if n is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Let p be the smallest odd prime divisor of n then a(n) = min(p,2n/p).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A111787; a
a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
sage: a.offset
1
sage: a(1)
0
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
5
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
_eval(n)

EXAMPLES:

sage: [sloane.A111787._eval(n) for n in range(1,11)]
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4]
_repr_()

EXAMPLES:

sage: sloane.A111787._repr_()
'a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.'
class sage.combinat.sloane_functions.ExponentialNumbers(a)
__init__(a)

A sequence of Exponential numbers.

EXAMPLES:

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0
_eval(n)

EXAMPLES:

sage: [sloane.A000110._eval(n) for n in range(10)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
_repr_()

EXAMPLES:

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(4)._repr_()
'Sequence of Exponential numbers around 4'
class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)
_eval(n)

EXAMPLES:

sage: [sloane.A000153._eval(n) for n in range(8)]
[0, 1, 2, 7, 32, 181, 1214, 9403]
_precompute(how_many=20)

EXAMPLES:

sage: sloane.A000153._precompute()
sage: v1 = len(sloane.A000153._b)
sage: sloane.A000153._precompute(10)
sage: len(sloane.A000153._b) - v1
10
gen(a0, a1, d)

EXAMPLES:

sage: it = sloane.A000153.gen(0,1,2)
sage: [it.next() for i in range(5)]
[0, 1, 2, 7, 32]
list(n)

EXAMPLES:

sage: sloane.A000153.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]
class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)
gen(a0, a1, d)

EXAMPLES:

sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
sage: e = ExtremesOfPermanentsSequence2()
sage: it = e.gen(6,43,6)
sage: [it.next() for i in range(5)]
[6, 43, 307, 2542, 23799]
class sage.combinat.sloane_functions.RecurrenceSequence(offset=1)
_eval(n)

EXAMPLES:

sage: [sloane.A001110._eval(n) for n in range(5)]
[0, 1, 36, 1225, 41616]
_precompute(how_many=20)

EXAMPLES:

sage: initial = len(sloane.A001110._b)
sage: sloane.A001110._precompute(10)
sage: len(sloane.A001110._b) - initial == 10
True
list(n)

EXAMPLES:

sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)
_eval(n)

EXAMPLES:

sage: [sloane.A001906._eval(n) for n in range(10)]
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
_precompute(how_many=150)

EXAMPLES:

sage: initial = len(sloane.A001906._b)
sage: sloane.A001906._precompute(10)
sage: len(sloane.A001906._b) - initial == 10
True
list(n)

EXAMPLES:

sage: sloane.A001906.list(10)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
class sage.combinat.sloane_functions.Sloane

A collection of Sloane generating functions.

This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with ‘A’. These are listed for tab completion, but not instantiated until requested.

EXAMPLES: Ensure we have lots of entries:

sage: len(sloane.trait_names()) > 100
True

And ensure none are being incorrectly returned:

sage: [ None for n in sloane.trait_names() if not n.startswith('A') ]
[]

Ensure we can access dynamic constructions and cache correctly:

sage: s = sloane.A000587
sage: s is sloane.A000587
True

And that we can access other functions in parent classes:

sage: sloane.__class__
<class 'sage.combinat.sloane_functions.Sloane'>

AUTHORS:

  • Nick Alexander
__getattribute__(name)

Construct and cache unique instances of Sloane generating function objects .

EXAMPLES:

sage: sloane.__getattribute__('A000001')
Number of groups of order n.
sage: sloane.__getattribute__('dog')
...
AttributeError: dog
__weakref__
list of weak references to the object (if defined)
trait_names()

List Sloane generating functions for tab-completion. The member classes are inspected from module sage.combinat.sloane_functions.

They must be sub classes of SloaneSequence and must start with ‘A’. These restrictions are only to prevent typos, incorrect inspecting, etc.

EXAMPLES:

sage: type(sloane.trait_names())
<type 'list'>
class sage.combinat.sloane_functions.SloaneSequence(offset=1)

Base class for a Sloane integer sequence.

EXAMPLES:

We create a dummy sequence:

__call__(n)

EXAMPLES:

sage: sloane.A000007(2)
0
sage: sloane.A000007('a')
...
TypeError: input must be an int, long, or Integer
sage: sloane.A000007(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: sloane.A000001(0)
...
ValueError: input n (=0) must be a positive integer
__cmp__(other)

EXAMPLES:

sage: cmp(sloane.A000007,sloane.A000045) == 0
False
sage: cmp(sloane.A000007,sloane.A000007) == 0
True
__getitem__(n)

Return sequence[n]. We interpret slices as best we can, but our sequences are infinite so we want to prevent some mis-incantations.

Therefore, we arbitrarily cap slices to be at most LENGTH=100000 elements long. Since many Sloane sequences are costly to compute, this is probably not an unreasonable decision, but just in case, list does not cap length.

EXAMPLES:

sage: sloane.A000012[3]
1
sage: sloane.A000012[:4]
[1, 1, 1, 1]
sage: sloane.A000012[:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: sloane.A000012[4:10]
[1, 1, 1, 1, 1, 1]
sage: sloane.A000012[0:1000000000]
...
IndexError: slice (=slice(0, 1000000000, None)) too long
__init__(offset=1)

A sequence starting at offset (=1 by default).

EXAMPLES:

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence().offset
1
sage: SloaneSequence(4).offset
4
__iter__()

EXAMPLES:

sage: iter(sloane.A000012)
...
NotImplementedError
__weakref__
list of weak references to the object (if defined)
_eval(n)

EXAMPLES:

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(0)._eval(4)
...
NotImplementedError
_repr_()

EXAMPLES:

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(4)._repr_()
...
NotImplementedError
_sage_src_()

Returns the source code for the class of self.

EXAMPLES:

sage: sloane.A000045._sage_src_()
'class A000045(...'
list(n)

Return n terms of the sequence: sequence[offset], sequence[offset+1], ... , sequence[offset+n-1]. EXAMPLES:

sage: sloane.A000012.list(4)
[1, 1, 1, 1]
sage.combinat.sloane_functions.perm_mh(m, h)

This functions calculates f(g,h) from Sloane’s sequences A079908-A079928

INPUT:

  • m - positive integer
  • h - non negative integer

OUTPUT: permanent of the m x (m+h) matrix, etc.

EXAMPLES:

sage: from sage.combinat.sloane_functions import perm_mh
sage: perm_mh(3,3)
36
sage: perm_mh(3,4)
76

AUTHORS:

  • Jaap Spies (2006)
sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)

homogeneous general second-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2)

EXAMPLES:

sage: from sage.combinat.sloane_functions import recur_gen2
sage: it = recur_gen2(1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)

inhomogenous second-order linear recurrence generator with fixed coefficients and b = f(n)

a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2) +f(n).

EXAMPLES:

sage: from sage.combinat.sloane_functions import recur_gen2b
sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)

homogeneous general third-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3)

EXAMPLES:

sage: from sage.combinat.sloane_functions import recur_gen3
sage: it = recur_gen3(1,1,1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

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