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)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
Traceback (most recent call last):
...
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

We agree with the online database::

sage: for t in sloane.trait_names():    # long time; optional -- internet
....:     online_list = list(oeis(t).first_terms())
....:     L = max(2, len(online_list) // 2)
....:     sage_list = sloane.__getattribute__(t).list(L)
....:     if online_list[:L] != sage_list:
....:         print t, 'seems wrong'

See also

  • If you want to get more informations relative to a sequence (references, links, examples, programs, ...), you can use the On-Line Encyclopedia of Integer Sequences provided by the OEIS module.
  • If you plan to do a lot of automatic searches for subsequences, you should consider installing SloaneEncyclopedia, a local partial copy of the OEIS.

AUTHORS:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000004

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000005

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000007

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000008

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000009

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000012

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000015

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000016

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000027

Bases: sage.combinat.sloane_functions.SloaneSequence

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]
class sage.combinat.sloane_functions.A000030

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000032

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000035

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000040

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-17)
class sage.combinat.sloane_functions.A000041

Bases: sage.combinat.sloane_functions.SloaneSequence

\(a(n)\) = 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)
class sage.combinat.sloane_functions.A000043

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000045

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000073

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000085

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000100

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000108

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000110

Bases: sage.combinat.sloane_functions.ExponentialNumbers

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
class sage.combinat.sloane_functions.A000120

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000129

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A000142

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000153

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-2)*a(n-2)\), 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)
class sage.combinat.sloane_functions.A000165

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000166

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000169

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000203

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A000204

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000213

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000225

Bases: sage.combinat.sloane_functions.SloaneSequence

\(2^n - 1\).

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000244

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000255

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-1)*a(n-2)\), 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)
class sage.combinat.sloane_functions.A000261

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-3)*a(n-2)\), 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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000272

Bases: sage.combinat.sloane_functions.SloaneSequence

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

INPUT:

  • n - 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)
1
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(12)
[1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

AUTHORS:

  • Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000290

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000292

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000302

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000312

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000326

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000330

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000396

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000578

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000583

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000587

Bases: sage.combinat.sloane_functions.ExponentialNumbers

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
class sage.combinat.sloane_functions.A000668

Bases: sage.combinat.sloane_functions.SloaneSequence

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

AUTHORS:

  • Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000670

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A000720

Bases: sage.combinat.sloane_functions.SloaneSequence

\(pi(n)\), 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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A000796

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

Prime powers

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A000961;a
Prime powers.
sage: a(0)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A001006

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A001045

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A001055

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
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

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

\(a(n)^2\) 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)
class sage.combinat.sloane_functions.A001110

Bases: sage.combinat.sloane_functions.RecurrenceSequence

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)
g(k)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A001157

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001189

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001221

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001222

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001227

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.A001333

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A001358

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A001477

Bases: sage.combinat.sloane_functions.SloaneSequence

The nonnegative integers.

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A001477;a
The nonnegative integers.
sage: a(-1)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001694

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

\(F(2n) =\) 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)
class sage.combinat.sloane_functions.A001909

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-4)*a(n-2)\), 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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001910

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-5)*a(n-2)\), 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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A001969

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A002110

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A002113

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A002378

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A002620

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A002808

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHOR:

  • Jaap Spies (2007-01-31)
class sage.combinat.sloane_functions.A004086

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A004526

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A005100

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

Square-free numbers

INPUT:

  • n - positive integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A005117;a
Square-free numbers.
sage: a(0)
Traceback (most recent call last):
...
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)
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A005843

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A006318

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A006530

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A006882

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A008275

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
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)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A010060

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A015521

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A015523

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A015530

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A015531

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A015551

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A018252

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A020639

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
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)

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A049310

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A051959

Bases: sage.combinat.sloane_functions.RecurrenceSequence

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)
g(k)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2

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

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)
class sage.combinat.sloane_functions.A061084

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
class sage.combinat.sloane_functions.A064553

Bases: sage.combinat.sloane_functions.SloaneSequence

\(a(1) = 1\), \(a(prime(i)) = i + 1\) 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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A079922(offset=1)

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.A079923(offset=1)

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-17)
class sage.combinat.sloane_functions.A082411

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A083103

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A083104

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A083105

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A083216

Bases: sage.combinat.sloane_functions.RecurrenceSequence2

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)
class sage.combinat.sloane_functions.A090010

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A090012

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A090013

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A090014

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A090015

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A090016

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
class sage.combinat.sloane_functions.A109814

Bases: sage.combinat.sloane_functions.SloaneSequence

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.

By definition, \(n\) 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 \(k = min(d,2n/d)\). \(a(n)\) can be alternatively defined as the largest among those \(k\) .

INPUT:

  • n - non negative integer

OUTPUT:

  • integer - function value

EXAMPLES:

sage: a = sloane.A109814; a
a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a.list(9)
[1, 1, 2, 1, 2, 3, 2, 1, 3]

AUTHORS:

  • Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A111774

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2007-01-13)
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

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

AUTHORS:

  • Jaap Spies (2006-12-09)
class sage.combinat.sloane_functions.A111787

Bases: sage.combinat.sloane_functions.SloaneSequence

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)
Traceback (most recent call last):
...
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)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

  • Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.ExponentialNumbers(a)

Bases: sage.combinat.sloane_functions.SloaneSequence

A sequence of Exponential numbers.

EXAMPLES:

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0
class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)

Bases: sage.combinat.sloane_functions.SloaneSequence

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
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)

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

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
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)

Bases: sage.combinat.sloane_functions.SloaneSequence

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
list(n)

EXAMPLES:

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

Bases: sage.combinat.sloane_functions.SloaneSequence

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
list(n)

EXAMPLES:

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

Bases: sage.structure.sage_object.SageObject

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
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)

Bases: sage.structure.sage_object.SageObject

Base class for a Sloane integer sequence.

EXAMPLES:

We create a dummy sequence:

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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