EXAMPLES:
Type sloane.[tab] to see a list of the sequences that are defined.
sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a(1)
1
sage: a(6)
4
sage: a(100)
9
Type d._eval?? to see how the function that computes an individual term of the sequence is implemented.
The input must be a positive integer:
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
You can also change how a sequence prints:
sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a.rename('(..., tau(n), ...)')
sage: a
(..., tau(n), ...)
sage: a.reset_name()
sage: a
The integer sequence tau(n), which is the number of divisors of n.
TESTS:
sage: a = sloane.A000001;
sage: a == loads(dumps(a))
True
AUTHORS:
Number of groups of order .
Note: The database_gap-4.4.9 must be installed for
.
run sage -i database_gap-4.4.9 or higher first.
INPUT:
OUTPUT: integer
EXAMPLES:
sage: a = sloane.A000001;a
Number of groups of order n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1) #optional database_gap
1
sage: a(2) #optional database_gap
1
sage: a(9) #optional database_gap
2
sage: a.list(16) #optional database_gap
[1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
sage: a(60) # optional
13
AUTHORS:
EXAMPLES:
sage: sloane.A000001._eval(4)
2
sage: sloane.A000001._eval(51) #optional requires database_gap
EXAMPLES:
sage: sloane.A000001._repr_()
'Number of groups of order n.'
The zero sequence.
INPUT:
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:
EXAMPLES:
sage: sloane.A000004._eval(5)
0
EXAMPLES:
sage: sloane.A000004._repr_()
'The zero sequence.'
The sequence , which is the number of divisors of
.
This sequence is also denoted (also called
or
), the number of
divisors of n.
INPUT:
OUTPUT:
EXAMPLES:
sage: d = sloane.A000005; d
The integer sequence tau(n), which is the number of divisors of n.
sage: d(1)
1
sage: d(6)
4
sage: d(51)
4
sage: d(100)
9
sage: d(0)
...
ValueError: input n (=0) must be a positive integer
sage: d.list(10)
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]
AUTHORS:
EXAMPLES:
sage: sloane.A000005._eval(5)
2
EXAMPLES:
sage: sloane.A000005._repr_()
'The integer sequence tau(n), which is the number of divisors of n.'
The characteristic function of 0: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000007._eval(n) for n in range(10)]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
EXAMPLES:
sage: sloane.A000007._repr_()
'The characteristic function of 0: a(n) = 0^n.'
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000008._eval(n) for n in range(14)]
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]
EXAMPLES:
sage: sloane.A000008._repr_()
'Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.'
Number of partitions of into odd parts.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000009._eval(i) for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
EXAMPLES:
sage: initial = len(sloane.A000009._b)
sage: sloane.A000009._precompute(10)
sage: len(sloane.A000009._b) - initial == 10
True
EXAMPLES:
sage: sloane.A000009._repr_()
'Number of partitions of n into odd parts.'
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]
EXAMPLES:
sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
The integer sequence A000010 is Euler’s totient function.
Number of positive integers that are relative prime
to
. Number of totatives of
.
Euler totient function : count numbers
and prime to
. euler_phi is a standard Sage function
implemented in PARI
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000010; a
Euler's totient function
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(11)
10
sage: a.list(12)
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000010._eval(n) for n in range(1,11)]
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4]
EXAMPLES:
sage: sloane.A000010._repr_()
"Euler's totient function"
The all 1’s sequence.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000012._eval(n) for n in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
EXAMPLES:
sage: sloane.A000012._repr_()
"The all 1's sequence."
Smallest prime power .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000015; a
Smallest prime power >= n.
sage: a(1)
1
sage: a(8)
8
sage: a(305)
307
sage: a(-4)
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000015._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11]
EXAMPLES:
sage: sloane.A000015._repr_()
'Smallest prime power >= n.'
Sloane’s A000016
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000016; a
Sloane's A000016.
sage: a(1)
1
sage: a(0)
1
sage: a(8)
16
sage: a(75)
251859545753048193000
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]
AUTHORS:
EXAMPLES:
sage: [sloane.A000016._eval(n) for n in range(10)]
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30]
EXAMPLES:
sage: sloane.A000016._repr_()
"Sloane's A000016."
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]
EXAMPLES:
sage: sloane.A000027._eval(5)
5
EXAMPLES:
sage: sloane.A000027._repr_()
'The natural numbers.'
Initial digit of .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000030; a
Initial digit of n
sage: a(0)
0
sage: a(1)
1
sage: a(8)
8
sage: a(454)
4
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]
AUTHORS:
EXAMPLES:
sage: [sloane.A000030._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
EXAMPLES:
sage: sloane.A000030._repr_()
'Initial digit of n'
Lucas numbers (beginning at 2): .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000032; a
Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
sage: a(0)
2
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]
AUTHORS:
EXAMPLES:
sage: [sloane.A000032._eval(n) for n in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
EXAMPLES:
sage: sloane.A000032._repr_()
'Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).'
A simple periodic sequence.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000035;a
A simple periodic sequence.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
0
sage: a(9)
1
sage: a.list(10)
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
AUTHORS:
EXAMPLES:
sage: [sloane.A000035._eval(n) for n in range(10)]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
EXAMPLES:
sage: sloane.A000035._repr_()
'A simple periodic sequence.'
The prime numbers.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000040; a
The prime numbers.
sage: a(1)
2
sage: a(8)
19
sage: a(305)
2011
sage: a.list(12)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000040._eval(n) for n in range(1,11)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
EXAMPLES:
sage: sloane.A000040._repr_()
'The prime numbers.'
= number of partitions of
(the partition
numbers).
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000041._eval(n) for n in range(1,11)]
[1, 2, 3, 5, 7, 11, 15, 22, 30, 42]
EXAMPLES:
sage: sloane.A000041._repr_()
'a(n) = number of partitions of n (the partition numbers).'
Primes such that
is prime.
is then called a Mersenne prime.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000043;a
Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
sage: a(1)
2
sage: a(2)
3
sage: a(39)
13466917
sage: a(40)
...
IndexError: list index out of range
sage: a.list(12)
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]
AUTHORS:
EXAMPLES:
sage: [sloane.A000043._eval(n) for n in range(1,11)]
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89]
EXAMPLES:
sage: sloane.A000043._repr_()
'Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.'
Sequence of Fibonacci numbers, offset 0,4.
REFERENCES:
We have one more. Our first Fibonacci number is 0.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000045; a
Fibonacci numbers with index n >= 0
sage: a(0)
0
sage: a(1)
1
sage: a.list(12)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000045._eval(n) for n in range(1,11)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
EXAMPLES:
sage: initial = len(sloane.A000045._b)
sage: sloane.A000045._precompute(10)
sage: len(sloane.A000045._b) - initial > 0
True
EXAMPLES:
sage: sloane.A000045._repr_()
'Fibonacci numbers with index n >= 0'
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]
EXAMPLES:
sage: sloane.A000045.list(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
Odious numbers: odd number of 1’s in binary expansion.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000069._eval(n) for n in range(10)]
[1, 2, 4, 7, 8, 11, 13, 14, 16, 19]
EXAMPLES:
sage: sloane.A000069._repr_()
"Odious numbers: odd number of 1's in binary expansion."
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000073._eval(n) for n in range(10)]
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
EXAMPLES:
sage: initial = len(sloane.A000073._b)
sage: sloane.A000073._precompute(10)
sage: len(sloane.A000073._b) - initial == 10
True
EXAMPLES:
sage: sloane.A000073._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
EXAMPLES:
sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
Powers of 2: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000079._eval(n) for n in range(10)]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
EXAMPLES:
sage: sloane.A000079._repr_()
'Powers of 2: a(n) = 2^n.'
Number of self-inverse permutations on letters, also
known as involutions; number of Young tableaux with
cells.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000085._eval(n) for n in range(10)]
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620]
EXAMPLES:
sage: sloane.A000085._repr_()
'Number of self-inverse permutations on n letters.'
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000100._eval(n) for n in range(10)]
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94]
EXAMPLES:
sage: sloane.A000100._repr_()
'Number of compositions of n in which the maximum part size is 3.'
Catalan numbers:
.
Also called Segner numbers.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000108._eval(n) for n in range(10)]
[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
EXAMPLES:
sage: sloane.A000108._repr_()
'Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.'
The sequence of Bell numbers.
The Bell number counts the number of ways to put
distinguishable things into indistinguishable boxes
such that no box is empty.
Let denote the Stirling number of the second
kind. Then
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000110._repr_()
'Sequence of Bell numbers'
1’s-counting sequence: number of 1’s in binary expansion of
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000120._eval(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
EXAMPLES:
sage: sloane.A000120._repr_()
"1's-counting sequence: number of 1's in binary expansion of n."
EXAMPLES:
sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
Central polygonal numbers (the Lazy Caterer’s sequence):
.
Or, maximal number of pieces formed when slicing a pancake with
cuts.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000124._eval(n) for n in range(10)]
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]
EXAMPLES:
sage: sloane.A000124._repr_()
"Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1."
Pell numbers: ,
; for
,
.
Denominators of continued fraction convergents to
.
See also A001333
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000129._repr_()
'Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).'
Factorial numbers:
Order of symmetric group , number of permutations of
letters.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000142._eval(n) for n in range(10)]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
EXAMPLES:
sage: sloane.A000142._repr_()
'Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).'
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000153._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.'
Double factorial numbers: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000165._eval(n) for n in range(10)]
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560]
EXAMPLES:
sage: sloane.A000165._repr_()
'Double factorial numbers: (2n)!! = 2^n*n!.'
Subfactorial or rencontres numbers, or derangements: number of
permutations of elements with no fixed points.
With offset 1 also the permanent of a (0,1)-matrix of order
with
0’s not on a line.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000166._eval(n) for n in range(9)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]
EXAMPLES:
sage: sloane.A000166._repr_()
'Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.'
Number of labeled rooted trees with nodes:
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000169;a
Number of labeled rooted trees with n nodes: n^(n-1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(10)
1000000000
sage: a.list(11)
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]
AUTHORS:
EXAMPLES:
sage: [sloane.A000169._eval(n) for n in range(1,11)]
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000]
EXAMPLES:
sage: sloane.A000169._repr_()
'Number of labeled rooted trees with n nodes: n^(n-1).'
The sequence , where
is the
sum of the divisors of
. Also called
.
The function sigma(n, k) implements
in Sage.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000203; a
sigma(n) = sum of divisors of n. Also called sigma_1(n).
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(256)
511
sage: a.list(12)
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000203._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
EXAMPLES:
sage: sloane.A000203._repr_()
'sigma(n) = sum of divisors of n. Also called sigma_1(n).'
Lucas numbers (beginning with 1):
with
,
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000204; a
Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000204._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
EXAMPLES:
sage: sloane.A000204._repr_()
'Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.'
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000213._eval(n) for n in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
EXAMPLES:
sage: initial = len(sloane.A000213._b)
sage: sloane.A000213._precompute(10)
sage: len(sloane.A000213._b) - initial == 10
True
EXAMPLES:
sage: sloane.A000213._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
EXAMPLES:
sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
Triangular numbers: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000217._eval(n) for n in range(10)]
[0, 1, 3, 6, 10, 15, 21, 28, 36, 45]
EXAMPLES:
sage: sloane.A000217._repr_()
'Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.'
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000225;a
2^n - 1.
sage: a(0)
0
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(12)
4095
sage: a.list(12)
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]
AUTHORS:
EXAMPLES:
sage: [sloane.A000225._eval(n) for n in range(10)]
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511]
EXAMPLES:
sage: sloane.A000225._repr_()
'2^n - 1.'
Powers of 3: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000244;a
Powers of 3: a(n) = 3^n.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(3)
27
sage: a(11)
177147
sage: a.list(12)
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]
AUTHORS:
EXAMPLES:
sage: [sloane.A000244._eval(n) for n in range(10)]
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
EXAMPLES:
sage: sloane.A000244._repr_()
'Powers of 3: a(n) = 3^n.'
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000255._repr_()
'a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.'
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000261;a
a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a.offset
1
sage: a(8)
30637
sage: a(22)
1801366114380914335441
sage: a.list(9)
[0, 1, 3, 13, 71, 465, 3539, 30637, 296967]
AUTHORS:
EXAMPLES:
sage: sloane.A000261._repr_()
'a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.'
Number of labeled rooted trees on nodes:
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(11)
[1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]
AUTHORS:
EXAMPLES:
sage: [sloane.A000272._eval(n) for n in range(1,11)]
[1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000]
EXAMPLES:
sage: sloane.A000272._repr_()
'Number of labeled rooted trees with n nodes: n^(n-2).'
The squares: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000290;a
The squares: a(n) = n^2.
sage: a(0)
0
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(16)
256
sage: a.list(17)
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]
AUTHORS:
EXAMPLES:
sage: [sloane.A000290._eval(n) for n in range(10)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
EXAMPLES:
sage: sloane.A000290._repr_()
'The squares: a(n) = n^2.'
Tetrahedral (or pyramidal) numbers:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000292._eval(n) for n in range(10)]
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165]
EXAMPLES:
sage: sloane.A000292._repr_()
'Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.'
Powers of 4: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000302._eval(n) for n in range(10)]
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]
EXAMPLES:
sage: sloane.A000302._repr_()
'Powers of 4: a(n) = 4^n.'
Number of labeled mappings from points to themselves
(endofunctions):
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000312;a
Number of labeled mappings from n points to themselves (endofunctions): n^n.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(1)
1
sage: a(9)
387420489
sage: a.list(11)
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]
AUTHORS:
EXAMPLES:
sage: [sloane.A000312._eval(n) for n in range(10)]
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489]
EXAMPLES:
sage: sloane.A000312._repr_()
'Number of labeled mappings from n points to themselves (endofunctions): n^n.'
Pentagonal numbers: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000326;a
Pentagonal numbers: n(3n-1)/2.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
5
sage: a(10)
145
sage: a.list(12)
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A000326._eval(n) for n in range(10)]
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117]
EXAMPLES:
sage: sloane.A000326._repr_()
'Pentagonal numbers: n(3n-1)/2.'
Square pyramidal numbers”
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000330;a
Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
14
sage: a(11)
506
sage: a.list(12)
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]
AUTHORS:
EXAMPLES:
sage: [sloane.A000330._eval(n) for n in range(10)]
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285]
EXAMPLES:
sage: sloane.A000330._repr_()
'Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.'
Perfect numbers: equal to sum of proper divisors.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000396;a
Perfect numbers: equal to sum of proper divisors.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
28
sage: a(7)
137438691328
sage: a.list(7)
[6, 28, 496, 8128, 33550336, 8589869056, 137438691328]
AUTHORS:
EXAMPLES:
sage: [sloane.A000396._eval(n) for n in range(1,6)]
[6, 28, 496, 8128, 33550336]
EXAMPLES:
sage: sloane.A000396._repr_()
'Perfect numbers: equal to sum of proper divisors.'
The cubes: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000578;a
The cubes: n^3
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
27
sage: a(11)
1331
sage: a.list(12)
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]
AUTHORS:
EXAMPLES:
sage: [sloane.A000578._eval(n) for n in range(10)]
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729]
EXAMPLES:
sage: sloane.A000578._repr_()
'The cubes: n^3'
Fourth powers: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000583;a
Fourth powers: n^4.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
16
sage: a(9)
6561
sage: a.list(10)
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]
AUTHORS:
EXAMPLES:
sage: [sloane.A000583._eval(n) for n in range(10)]
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]
EXAMPLES:
sage: sloane.A000583._repr_()
'Fourth powers: n^4.'
The sequence of Uppuluri-Carpenter numbers.
The Uppuluri-Carpenter number counts the imbalance
in the number of ways to put
distinguishable things
into an even number of indistinguishable boxes versus into an odd
number of indistinguishable boxes, such that no box is empty.
Let denote the Stirling number of the second
kind. Then
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000587._repr_()
'Sequence of Uppuluri-Carpenter numbers'
Mersenne primes (of form where
is a
prime).
(See A000043 for the values of .)
Warning: a(39) has 4,053,946 digits!
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000668;a
Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
sage: a(1)
3
sage: a(2)
7
sage: a(12)
170141183460469231731687303715884105727
Warning: a(39) has 4,053,946 digits!
sage: a(40)
...
IndexError: list index out of range
sage: a.list(8)
[3, 7, 31, 127, 8191, 131071, 524287, 2147483647]
AUTHORS:
EXAMPLES:
sage: [sloane.A000668._eval(n) for n in range(1,11)]
[3,
7,
31,
127,
8191,
131071,
524287,
2147483647,
2305843009213693951,
618970019642690137449562111]
EXAMPLES:
sage: sloane.A000668._repr_()
'Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)'
Number of preferential arrangements of labeled
elements; or number of weak orders on
labeled
elements.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000670._eval(n) for n in range(1,10)]
[1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
EXAMPLES:
sage: sloane.A000670._repr_()
'Number of preferential arrangements of n labeled elements.'
, the number of primes
. Sometimes
called
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000720;a
pi(n), the number of primes <= n. Sometimes called PrimePi(n)
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a(8)
4
sage: a(1000)
168
sage: a.list(12)
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
AUTHORS:
EXAMPLES:
sage: [sloane.A000720._eval(n) for n in range(1,11)]
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4]
EXAMPLES:
sage: sloane.A000720._repr_()
'pi(n), the number of primes <= n. Sometimes called PrimePi(n)'
Decimal expansion of .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000796;a
Decimal expansion of Pi.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(13)
9
sage: a.list(14)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
sage: a(100)
7
AUTHOR:
EXAMPLES:
sage: [sloane.A000796._eval(n) for n in range(1,11)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
EXAMPLES:
sage: initial = len(sloane.A000796._b)
sage: sloane.A000796._precompute(10)
sage: len(sloane.A000796._b) - initial
10
EXAMPLES:
sage: sloane.A000796._repr_()
'Decimal expansion of Pi.'
EXAMPLES:
sage: sloane.A000796.list(10)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
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]
Prime powers
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A000961;a
Prime powers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]
AUTHORS:
EXAMPLES:
sage: [sloane.A000961._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
EXAMPLES:
sage: initial = len(sloane.A000961._b)
sage: sloane.A000961._precompute()
sage: len(sloane.A000961._b) - initial > 0
True
EXAMPLES:
sage: sloane.A000961._repr_()
'Prime powers.'
EXAMPLES:
sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
Central binomial coefficients:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000984._eval(n) for n in range(10)]
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
EXAMPLES:
sage: sloane.A000984._repr_()
'Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2'
Motzkin numbers: number of ways of drawing any number of
nonintersecting chords among points on a circle.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A001006._eval(n) for n in range(10)]
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
EXAMPLES:
sage: sloane.A001006._repr_()
'Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.'
Jacobsthal sequence: ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001045._repr_()
'Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).'
Number of ways of factoring with all factors 1.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001055;a
Number of ways of factoring n with all factors >1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(9)
2
sage: a.list(16)
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]
AUTHORS:
EXAMPLES:
sage: [sloane.A001055._eval(n) for n in range(1,11)]
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2]
EXAMPLES:
sage: sloane.A001055._repr_()
'Number of ways of factoring n with all factors >1.'
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
is a triangular number:
with
,
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001109._repr_()
'a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1'
Numbers that are both triangular and square:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001110._repr_()
'Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.'
EXAMPLES:
sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0
Double factorial numbers:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A001147._eval(n) for n in range(10)]
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
EXAMPLES:
sage: sloane.A001147._repr_()
'Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).'
The sequence , sum of squares of divisors of
.
The function sigma(n, k) implements in Sage.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001157;a
sigma_2(n): sum of squares of divisors of n
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
5
sage: a(8)
85
sage: a.list(9)
[1, 5, 10, 21, 26, 50, 50, 85, 91]
AUTHORS:
EXAMPLES:
sage: [sloane.A001157._eval(n) for n in range(1,11)]
[1, 5, 10, 21, 26, 50, 50, 85, 91, 130]
EXAMPLES:
sage: sloane.A001157._repr_()
'sigma_2(n): sum of squares of divisors of n'
Number of degree-n permutations of order exactly 2.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001189;a
Number of degree-n permutations of order exactly 2.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(2)
1
sage: a(12)
140151
sage: a.list(13)
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]
AUTHORS:
EXAMPLES:
sage: [sloane.A001189._eval(n) for n in range(1,11)]
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495]
EXAMPLES:
sage: sloane.A001189._repr_()
'Number of degree-n permutations of order exactly 2.'
Number of different prime divisors of
Also called omega(n) or . Maximal number of
terms in any factorization of
. Number of prime powers
that divide
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001221; a
Number of distinct primes dividing n (also called omega(n)).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
1
sage: a(41)
1
sage: a(84792)
3
sage: a.list(12)
[0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]
AUTHORS:
EXAMPLES:
sage: [sloane.A001221._eval(n) for n in range(1,10)]
[0, 1, 1, 1, 1, 2, 1, 1, 1]
EXAMPLES:
sage: sloane.A001221._repr_()
'Number of distinct primes dividing n (also called omega(n)).'
Number of prime divisors of (counted with
multiplicity).
Also called bigomega(n) or . Maximal number of
terms in any factorization of
. Number of prime powers
that divide
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001222; a
Number of prime divisors of n (counted with multiplicity).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
3
sage: a(41)
1
sage: a(84792)
5
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]
AUTHORS:
EXAMPLES:
sage: [sloane.A001222._eval(n) for n in range(1,10)]
[0, 1, 1, 2, 1, 2, 1, 3, 2]
EXAMPLES:
sage: sloane.A001222._repr_()
'Number of prime divisors of n (counted with multiplicity).'
Number of odd divisors of .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001227; a
Number of odd divisors of n
sage: a.offset
1
sage: a(1)
1
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
3
sage: a(256)
1
sage: a(29)
2
sage: a.list(20)
[1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A001227._eval(n) for n in range(1,10)]
[1, 1, 2, 1, 2, 2, 2, 1, 3]
EXAMPLES:
sage: sloane.A001227._repr_()
'Number of odd divisors of n'
Numerators of continued fraction convergents to .
See also A000129
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001333._repr_()
'Numerators of continued fraction convergents to sqrt(2).'
Products of two primes.
These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001358;a
Products of two primes.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(8)
22
sage: a(200)
669
sage: a.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]
AUTHORS:
EXAMPLES:
sage: [sloane.A001358._eval(n) for n in range(1,10)]
[4, 6, 9, 10, 14, 15, 21, 22, 25]
EXAMPLES:
sage: initial = len(sloane.A001358._b)
sage: sloane.A001358._precompute()
sage: len(sloane.A001358._b) - initial > 0
True
EXAMPLES:
sage: sloane.A001358._repr_()
'Products of two primes.'
EXAMPLES:
sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]
Central binomial coefficients:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A001405._eval(n) for n in range(10)]
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
EXAMPLES:
sage: sloane.A001405._repr_()
'Central binomial coefficients: C(n,floor(n/2)).'
The nonnegative integers.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001477;a
The nonnegative integers.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3382789)
3382789
sage: a(11)
11
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
AUTHORS:
EXAMPLES:
sage: [sloane.A001477._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
EXAMPLES:
sage: sloane.A001477._repr_()
'The nonnegative integers.'
This function returns the -th Powerful Number:
A positive integer is powerful if for every prime
dividing
,
also divides
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001694; a
Powerful Numbers (also called squarefull, square-full or 2-full numbers).
sage: a.offset
1
sage: a(1)
1
sage: a(4)
9
sage: a(100)
3136
sage: a(156)
7225
sage: a.list(19)
[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A001694._eval(n) for n in range(1,10)]
[1, 4, 8, 9, 16, 25, 27, 32, 36]
EXAMPLES:
sage: sloane.A001694._powerful_numbers_in_range(0,50)
[4, 8, 9, 16, 25, 27, 32, 36, 49]
EXAMPLES:
sage: initial = len(sloane.A001694._b)
sage: sloane.A001694._precompute()
sage: len(sloane.A001694._b) - initial > 0
True
EXAMPLES:
sage: sloane.A001694._repr_()
'Powerful Numbers (also called squarefull, square-full or 2-full numbers).'
This function returns True if and only if is a Powerful
Number:
A positive integer is powerful if for every prime
dividing
,
also divides
. See Sloane’s OEIS A001694.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001694
sage: a.is_powerful(2500)
True
sage: a.is_powerful(20)
False
AUTHORS:
EXAMPLES:
sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]
Numbers such that
,
where
is Euler’s totient function.
Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001836; a
Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
sage: a.offset
1
sage: a(1)
53
sage: a(8)
683
sage: a(300)
17798
sage: a.list(12)
[53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
Compare: Searching Sloane’s online database... Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
AUTHORS:
EXAMPLES:
sage: [sloane.A001836._eval(n) for n in range(1,10)]
[53, 83, 158, 263, 293, 368, 578, 683, 743]
EXAMPLES:
sage: initial = len(sloane.A001836._b)
sage: sloane.A001836._precompute()
sage: len(sloane.A001836._b) - initial > 0
True
EXAMPLES:
sage: sloane.A001836._repr_()
"Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010."
EXAMPLES:
sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]
bisection of Fibonacci sequence:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001906._repr_()
'F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).'
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001909;a
a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
sage: a(1)
...
ValueError: input n (=1) must be an integer >= 2
sage: a.offset
2
sage: a(2)
0
sage: a(8)
8544
sage: a(22)
470033715095287415734
sage: a.list(9)
[0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]
AUTHORS:
EXAMPLES:
sage: sloane.A001909._repr_()
'a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.'
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001910;a
a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
sage: a(0)
...
ValueError: input n (=0) must be an integer >= 3
sage: a(3)
0
sage: a.offset
3
sage: a(8)
1909
sage: a(22)
98125321641110663023
sage: a.list(9)
[0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]
AUTHORS:
EXAMPLES:
sage: sloane.A001910._repr_()
'a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.'
Evil numbers: even number of 1’s in binary expansion.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A001969._eval(n) for n in range(10)]
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18]
EXAMPLES:
sage: sloane.A001969._repr_()
"Evil numbers: even number of 1's in binary expansion."
Primorial numbers (first definition): product of first
primes. Sometimes written
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A002110._eval(n) for n in range(10)]
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870]
EXAMPLES:
sage: sloane.A002110._repr_()
'Primorial numbers (first definition): product of first n primes. Sometimes written p#.'
Palindromes in base 10.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A002113._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
EXAMPLES:
sage: initial = len(sloane.A002113._b)
sage: sloane.A002113._precompute()
sage: len(sloane.A002113._b) - initial > 0
True
EXAMPLES:
sage: sloane.A002113._repr_()
'Palindromes in base 10.'
EXAMPLES:
sage: sloane.A002113.list(15)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
Repunits: . Often denoted by
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A002275._eval(n) for n in range(10)]
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111]
EXAMPLES:
sage: sloane.A002275._repr_()
'Repunits: (10^n - 1)/9. Often denoted by R_n.'
Oblong (or pronic, or heteromecic) numbers: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A002378;a
Oblong (or pronic, or heteromecic) numbers: n(n+1).
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(1)
2
sage: a(11)
132
sage: a.list(12)
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]
AUTHORS:
EXAMPLES:
sage: [sloane.A002378._eval(n) for n in range(10)]
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90]
EXAMPLES:
sage: sloane.A002378._repr_()
'Oblong (or pronic, or heteromecic) numbers: n(n+1).'
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently,
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A002620._eval(n) for n in range(10)]
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20]
EXAMPLES:
sage: sloane.A002620._repr_()
'Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).'
The composite numbers: numbers of the form
for
and
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A002808;a
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(11)
20
sage: a.list(12)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]
AUTHORS:
EXAMPLES:
sage: [sloane.A002808._eval(n) for n in range(1,11)]
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
EXAMPLES:
sage: initial = len(sloane.A002808._b)
sage: sloane.A002808._precompute()
sage: len(sloane.A002808._b) - initial > 0
True
EXAMPLES:
sage: sloane.A002808._repr_()
'The composite numbers: numbers n of the form x*y for x > 1 and y > 1.'
EXAMPLES:
sage: sloane.A002808.list(10)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
Least common multiple (or lcm) of .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A003418;a
Least common multiple (or lcm) of {1, 2, ..., n}.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
360360
sage: a.list(14)
[1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
sage: a(20.0)
...
TypeError: input must be an int, long, or Integer
AUTHOR:
EXAMPLES:
sage: [sloane.A003418._eval(n) for n in range(1,11)]
[1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520]
EXAMPLES:
sage: sloane.A003418._repr_()
'Least common multiple (or lcm) of {1, 2, ..., n}.'
Read n backwards (referred to as in many
sequences).
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A004086._eval(n) for n in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
EXAMPLES:
sage: sloane.A004086._repr_()
'Read n backwards (referred to as R(n) in many sequences).'
The nonnegative integers repeated
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A004526._eval(n) for n in range(10)]
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4]
EXAMPLES:
sage: sloane.A004526._repr_()
'The nonnegative integers repeated.'
Deficient numbers: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A005100;a
Deficient numbers: sigma(n) < 2n
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(12)
14
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]
AUTHORS:
EXAMPLES:
sage: [sloane.A005100._eval(n) for n in range(1,10)]
[1, 2, 3, 4, 5, 7, 8, 9, 10]
EXAMPLES:
sage: initial = len(sloane.A005100._b)
sage: sloane.A005100._precompute()
sage: len(sloane.A005100._b) - initial > 0
True
EXAMPLES:
sage: sloane.A005100._repr_()
'Deficient numbers: sigma(n) < 2n'
EXAMPLES:
sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
Abundant numbers (sum of divisors of exceeds
).
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A005101;a
Abundant numbers (sum of divisors of n exceeds 2n).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
12
sage: a(2)
18
sage: a(12)
60
sage: a.list(12)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]
AUTHORS:
EXAMPLES:
sage: [sloane.A005101._eval(n) for n in range(1,11)]
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
EXAMPLES:
sage: initial = len(sloane.A005101._b)
sage: sloane.A005101._precompute()
sage: len(sloane.A005101._b) - initial > 0
True
EXAMPLES:
sage: sloane.A005101._repr_()
'Abundant numbers (sum of divisors of n exceeds 2n).'
EXAMPLES:
sage: sloane.A005101.list(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
Square-free numbers
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A005117;a
Square-free numbers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]
AUTHORS:
EXAMPLES:
sage: [sloane.A005117._eval(n) for n in range(1,11)]
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
EXAMPLES:
sage: initial = len(sloane.A005117._b)
sage: sloane.A005117._precompute()
sage: len(sloane.A005117._b) - initial > 0
True
EXAMPLES:
sage: sloane.A005117._repr_()
'Square-free numbers.'
EXAMPLES:
sage: sloane.A005117.list(10)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
The odd numbers a(n) = 2n + 1.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A005408;a
The odd numbers a(n) = 2n + 1.
sage: a(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(4)
9
sage: a(11)
23
sage: a.list(12)
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]
AUTHORS:
EXAMPLES:
sage: [sloane.A005408._eval(n) for n in range(10)]
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19]
EXAMPLES:
sage: sloane.A005408._repr_()
'The odd numbers a(n) = 2n + 1.'
The even numbers: .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A005843;a
The even numbers: a(n) = 2n.
sage: a(0.0)
...
TypeError: input must be an int, long, or Integer
sage: a(1)
2
sage: a(2)
4
sage: a(9)
18
sage: a.list(10)
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
AUTHORS:
EXAMPLES:
sage: [sloane.A005843._eval(n) for n in range(10)]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
EXAMPLES:
sage: sloane.A005843._repr_()
'The even numbers: a(n) = 2n.'
Large Schroeder numbers.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A006318._eval(n) for n in range(10)]
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]
EXAMPLES:
sage: sloane.A006318._repr_()
'Large Schroeder numbers.'
Largest prime dividing (with
).
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A006530;a
Largest prime dividing n (with a(1)=1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(8)
2
sage: a(11)
11
sage: a.list(15)
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]
AUTHORS:
EXAMPLES:
sage: [sloane.A006530._eval(n) for n in range(1,11)]
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5]
EXAMPLES:
sage: sloane.A006530._repr_()
'Largest prime dividing n (with a(1)=1).'
Double factorials :
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A006882._eval(n) for n in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
EXAMPLES:
sage: initial = len(sloane.A006882._b)
sage: sloane.A006882._precompute(10)
sage: len(sloane.A006882._b) - initial == 10
True
EXAMPLES:
sage: sloane.A006882._repr_()
'Double factorials n!!: a(n)=n*a(n-2).'
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]
EXAMPLES:
sage: sloane.A006882.list(10)
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
Pascal’s triangle read by rows:
,
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A007318._eval(n) for n in range(10)]
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
EXAMPLES:
sage: sloane.A007318._repr_()
"Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n."
Triangle of Stirling numbers of first kind, ,
,
.
The unsigned numbers are also called Stirling cycle numbers:
= number of permutations of
objects
with exactly
cycles.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A008275;a
Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
-1
sage: a(3)
1
sage: a(11)
24
sage: a.list(12)
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]
AUTHORS:
EXAMPLES:
sage: [sloane.A008275._eval(n) for n in range(1, 11)]
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1]
EXAMPLES:
sage: sloane.A008275._repr_()
'Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.'
EXAMPLES:
sage: sloane.A008275.s(4,2)
11
sage: sloane.A008275.s(5,2)
-50
sage: sloane.A008275.s(5,3)
35
Triangle of Stirling numbers of 2nd kind, ,
,
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A008277;a
Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(3)
1
sage: a(4.0)
...
TypeError: input must be an int, long, or Integer
sage: a.list(15)
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]
AUTHORS:
EXAMPLES:
sage: [sloane.A008277._eval(n) for n in range(1,11)]
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1]
EXAMPLES:
sage: sloane.A008277._repr_()
'Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.'
Returns the Stirling number S2(n,k) of the 2nd kind.
EXAMPLES:
sage: sloane.A008277.s2(4,2)
7
Moebius function .
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A008683;a
Moebius function mu(n).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
-1
sage: a(12)
0
sage: a.list(12)
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]
AUTHORS:
EXAMPLES:
sage: [sloane.A008683._eval(n) for n in range(1,11)]
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1]
EXAMPLES:
sage: sloane.A008683._repr_()
'Moebius function mu(n).'
Thue-Morse sequence.
Let denote the first
terms; then
, and for
,
, where
is obtained
from
by interchanging 0’s and 1’s.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A010060._eval(n) for n in range(10)]
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0]
EXAMPLES:
sage: sloane.A010060._repr_()
'Thue-Morse sequence.'
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A015521._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).'
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A015523._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).'
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A015530._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).'
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A015531._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).'
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A015551._repr_()
'Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).'
The nonprime numbers, starting with 1.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A018252;a
The nonprime numbers.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
4
sage: a(9)
15
sage: a.list(10)
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]
AUTHORS:
EXAMPLES:
sage: [sloane.A018252._eval(n) for n in range(1,11)]
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]
EXAMPLES:
sage: sloane.A018252._repr_()
'The nonprime numbers.'
Least prime dividing with
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A020639;a
Least prime dividing n (a(1)=1).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(13)
13
sage: a.list(14)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]
AUTHORS:
EXAMPLES:
sage: [sloane.A020639._eval(n) for n in range(1,11)]
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
EXAMPLES:
sage: initial = len(sloane.A020639._b)
sage: sloane.A020639._precompute(10)
sage: len(sloane.A020639._b) - initial == 10
True
EXAMPLES:
sage: sloane.A020639._repr_()
'Least prime dividing n (a(1)=1).'
EXAMPLES:
sage: sloane.A020639.list(10)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
Excess of = number of prime divisors (with
multiplicity) - number of prime divisors (without multiplicity).
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
2
sage: a(41)
0
sage: a(84792)
2
sage: a.list(12)
[0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]
AUTHORS:
- Jaap Spies (2007-01-19)
EXAMPLES:
sage: [sloane.A046660._eval(n) for n in range(1,10)]
[0, 0, 0, 1, 0, 0, 0, 2, 1]
EXAMPLES:
sage: sloane.A046660._repr_()
'Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).'
Triangle of coefficients of Chebyshev’s :
polynomials (exponents in increasing
order).
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A049310._eval(n) for n in range(10)]
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1]
EXAMPLES:
sage: sloane.A049310._repr_()
"Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order)."
Linear second order recurrence. A051959.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A051959._repr_()
'Linear second order recurrence. A051959.'
EXAMPLES:
sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0
.
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:
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A055790._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].'
Fibonacci-type sequence based on subtraction: ,
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A061084._eval(n) for n in range(10)]
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47]
EXAMPLES:
sage: sloane.A061084._repr_()
'Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).'
,
for
and
for
.
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A064553;a
a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(9)
9
sage: a.list(16)
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]
AUTHORS:
EXAMPLES:
sage: [sloane.A064553._eval(n) for n in range(1,11)]
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8]
EXAMPLES:
sage: sloane.A064553._repr_()
'a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0'
function returns solutions to the Dancing School problem with
girls and
boys.
The value is , the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A079922; a
Solutions to the Dancing School problem with n girls and n+3 boys
sage: a.offset
1
sage: a(1)
4
sage: a(8)
2227
sage: a.list(8)
[4, 13, 36, 90, 212, 478, 1044, 2227]
Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer
AUTHORS:
- Jaap Spies (2007-01-14)
EXAMPLES:
sage: [sloane.A079922._eval(n) for n in range(1,5)]
[4, 13, 36, 90]
EXAMPLES:
sage: sloane.A079922._repr_()
'Solutions to the Dancing School problem with n girls and n+3 boys'
function returns solutions to the Dancing School problem with
girls and
boys.
The value is , the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A079923; a
Solutions to the Dancing School problem with n girls and n+4 boys
sage: a.offset
1
sage: a(1)
5
sage: a(8)
15458
sage: a.list(8)
[5, 21, 76, 246, 738, 2108, 5794, 15458]
Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
AUTHORS:
- Jaap Spies (2007-01-17)
EXAMPLES:
sage: [sloane.A079923._eval(n) for n in range(1,11)]
[5, 21, 76, 246, 738, 2108, 5794, 15458, 40296, 103129]
EXAMPLES:
sage: sloane.A079923._repr_()
'Solutions to the Dancing School problem with n girls and n+4 boys'
Second-order linear recurrence sequence with
.
,
. This
is the second-order linear recurrence sequence with
and
co-prime, that R. L. Graham in 1964 stated did
not contain any primes.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A082411._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Second-order linear recurrence sequence with
.
,
. This is the
second-order linear recurrence sequence with
and
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:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A083103._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Second-order linear recurrence sequence with
.
,
. This is the
second-order linear recurrence sequence with
and
co-prime. It was found by Ronald Graham in 1990.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A083104._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Second-order linear recurrence sequence with
.
,
. This is the second-order linear
recurrence sequence with
and
co-prime. It was found by Donald Knuth in 1990.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A083105._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Second-order linear recurrence sequence with
.
,
. This is a second-order linear
recurrence sequence with
and
co-prime
that does not contain any primes. It was found by Herbert Wilf in
1990.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A083216._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090010;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
43
sage: a.offset
1
sage: a(8)
67741129
sage: a(22)
192416593029158989003270143
sage: a.list(9)
[6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]
AUTHORS:
EXAMPLES:
sage: sloane.A090010._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.'
Permanent of (0,1)-matrix of size with
and
zeros not on a line.
,
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090012;a
Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(2)
9
sage: a.offset
1
sage: a(8)
890901
sage: a(22)
129020386652297208795129
sage: a.list(9)
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]
AUTHORS:
EXAMPLES:
sage: [sloane.A090012._eval(n) for n in range(1,11)]
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465]
EXAMPLES:
sage: sloane.A090012._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.'
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090013;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
4
sage: a(2)
16
sage: a.offset
1
sage: a(8)
3481096
sage: a(22)
1112998577171142607670336
sage: a.list(9)
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]
AUTHORS:
EXAMPLES:
sage: [sloane.A090013._eval(n) for n in range(1,11)]
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284, 514872176]
EXAMPLES:
sage: sloane.A090013._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090014;a
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
5
sage: a(2)
25
sage: a.offset
1
sage: a(8)
11016595
sage: a(22)
7469733600354446865509725
sage: a.list(9)
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]
AUTHORS:
EXAMPLES:
sage: [sloane.A090014._eval(n) for n in range(1,11)]
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365, 1893369505]
EXAMPLES:
sage: sloane.A090014._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.'
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090015;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
36
sage: a.offset
1
sage: a(8)
29976192
sage: a(22)
41552258517692116794936876
sage: a.list(9)
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
AUTHORS:
EXAMPLES:
sage: [sloane.A090015._eval(n) for n in range(1,10)]
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
EXAMPLES:
sage: sloane.A090015._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A090016;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
7
sage: a(2)
49
sage: a.offset
1
sage: a(8)
72737161
sage: a(22)
199341969448774341802426289
sage: a.list(9)
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]
AUTHORS:
EXAMPLES:
sage: [sloane.A090016._eval(n) for n in range(1,10)]
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]
EXAMPLES:
sage: sloane.A090016._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.'
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
consecutive integers (other than the trivial
for
).
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A111774; a
Numbers that can be written as a sum of at least three consecutive positive integers.
sage: a(1)
6
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
141
sage: a(156)
209
sage: a(302)
386
sage: a.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A111774._eval(n) for n in range(1,11)]
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22]
EXAMPLES:
sage: initial = len(sloane.A111774._b)
sage: sloane.A111774._precompute()
sage: len(sloane.A111774._b) - initial > 0
True
EXAMPLES:
sage: sloane.A111774._repr_()
'Numbers that can be written as a sum of at least three consecutive positive integers.'
This function returns True if and only if 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 consecutive integers
(other than the trivial
for
).
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
Number of ways 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. strongly depends
on the number of odd divisors of
(A001227): Suppose
is to be written as sum of
consecutive
integers starting with
, then
. Only one of the factors is odd. For
each odd divisor of
there is a unique corresponding
,
and
must be excluded.
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A111775; a
Number of ways n can be written as a sum of at least three consecutive integers.
sage: a(1)
0
sage: a(0)
0
We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.
sage: a(15)
2
sage: a(100)
2
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer
AUTHORS:
EXAMPLES:
sage: [sloane.A111775._eval(n) for n in range(10)]
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
EXAMPLES:
sage: sloane.A111775._repr_()
'Number of ways n can be written as a sum of at least three consecutive integers.'
The is the
largest
such that
can be written as sum of
consecutive integers.
is the sum of at most
consecutive
positive integers. Suppose
is to be written as sum of
consecutive integers starting with
, then
. Only one of the factors is odd. For
each odd divisor
of
there is a unique
corresponding
.
is the
largest among those
. See:
http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A111776; a
a(n) is the largest k such that n can be written as sum of k consecutive integers.
sage: a(0)
1
sage: a(2)
1
sage: a.list(9)
[1, 1, 1, 2, 1, 2, 3, 2, 1]
AUTHORS:
EXAMPLES:
sage: [sloane.A111776._eval(n) for n in range(10)]
[1, 1, 1, 2, 1, 2, 3, 2, 1, 3]
EXAMPLES:
sage: sloane.A111776._repr_()
'a(n) is the largest k such that n can be written as sum of k consecutive integers.'
This function returns the -th number of Sloane’s
sequence A111787
if
is an odd prime or a power of 2.
For numbers of the third kind (see A111774) we proceed as follows:
suppose
is to be written as sum of
consecutive integers starting with
, then
. Let
be the smallest odd
prime divisor of
then
.
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
INPUT:
OUTPUT:
EXAMPLES:
sage: a = sloane.A111787; a
a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
sage: a.offset
1
sage: a(1)
0
sage: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
5
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
sage: a(-1)
...
ValueError: input n (=-1) must be a positive integer
AUTHORS:
EXAMPLES:
sage: [sloane.A111787._eval(n) for n in range(1,11)]
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4]
EXAMPLES:
sage: sloane.A111787._repr_()
'a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.'
A sequence of Exponential numbers.
EXAMPLES:
sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0
EXAMPLES:
sage: [sloane.A000110._eval(n) for n in range(10)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
EXAMPLES:
sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(4)._repr_()
'Sequence of Exponential numbers around 4'
EXAMPLES:
sage: [sloane.A000153._eval(n) for n in range(8)]
[0, 1, 2, 7, 32, 181, 1214, 9403]
EXAMPLES:
sage: sloane.A000153._precompute()
sage: v1 = len(sloane.A000153._b)
sage: sloane.A000153._precompute(10)
sage: len(sloane.A000153._b) - v1
10
EXAMPLES:
sage: it = sloane.A000153.gen(0,1,2)
sage: [it.next() for i in range(5)]
[0, 1, 2, 7, 32]
EXAMPLES:
sage: sloane.A000153.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]
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]
EXAMPLES:
sage: [sloane.A001110._eval(n) for n in range(5)]
[0, 1, 36, 1225, 41616]
EXAMPLES:
sage: initial = len(sloane.A001110._b)
sage: sloane.A001110._precompute(10)
sage: len(sloane.A001110._b) - initial == 10
True
EXAMPLES:
sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
EXAMPLES:
sage: [sloane.A001906._eval(n) for n in range(10)]
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
EXAMPLES:
sage: initial = len(sloane.A001906._b)
sage: sloane.A001906._precompute(10)
sage: len(sloane.A001906._b) - initial == 10
True
EXAMPLES:
sage: sloane.A001906.list(10)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
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:
Construct and cache unique instances of Sloane generating function objects .
EXAMPLES:
sage: sloane.__getattribute__('A000001')
Number of groups of order n.
sage: sloane.__getattribute__('dog')
...
AttributeError: dog
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'>
Base class for a Sloane integer sequence.
EXAMPLES:
We create a dummy sequence:
EXAMPLES:
sage: sloane.A000007(2)
0
sage: sloane.A000007('a')
...
TypeError: input must be an int, long, or Integer
sage: sloane.A000007(-1)
...
ValueError: input n (=-1) must be an integer >= 0
sage: sloane.A000001(0)
...
ValueError: input n (=0) must be a positive integer
EXAMPLES:
sage: cmp(sloane.A000007,sloane.A000045) == 0
False
sage: cmp(sloane.A000007,sloane.A000007) == 0
True
Return sequence[n]. We interpret slices as best we can, but our sequences are infinite so we want to prevent some mis-incantations.
Therefore, we arbitrarily cap slices to be at most LENGTH=100000 elements long. Since many Sloane sequences are costly to compute, this is probably not an unreasonable decision, but just in case, list does not cap length.
EXAMPLES:
sage: sloane.A000012[3]
1
sage: sloane.A000012[:4]
[1, 1, 1, 1]
sage: sloane.A000012[:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: sloane.A000012[4:10]
[1, 1, 1, 1, 1, 1]
sage: sloane.A000012[0:1000000000]
...
IndexError: slice (=slice(0, 1000000000, None)) too long
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
EXAMPLES:
sage: iter(sloane.A000012)
...
NotImplementedError
EXAMPLES:
sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(0)._eval(4)
...
NotImplementedError
EXAMPLES:
sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(4)._repr_()
...
NotImplementedError
Returns the source code for the class of self.
EXAMPLES:
sage: sloane.A000045._sage_src_()
'class A000045(...'
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]
This functions calculates from Sloane’s sequences
A079908-A079928
INPUT:
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:
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]
inhomogenous second-order linear recurrence generator with fixed
coefficients and
,
,
.
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]
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]