Safe Haskell	Safe-Inferred
Language	Haskell2010

Music.Theory.Math.Oeis

Description

The On-Line Encyclopedia of Integer Sequences, http://oeis.org/

Synopsis

a000005 :: Integral n => [n]
a000010 :: Integral n => [n]
a000010_n :: Integral n => n -> n
a000012 :: Num n => [n]
a000031 :: Integral n => [n]
a000031_n :: Integral n => n -> n
a000032 :: Num n => [n]
a000040 :: Integral n => [n]
a000041 :: Num n => [n]
a000045 :: Num n => [n]
a000051 :: Num n => [n]
a000071 :: Num n => [n]
a000073 :: Num n => [n]
a000078 :: Num n => [n]
a000079 :: Num n => [n]
a000085 :: Integral n => [n]
a000108 :: Num n => [n]
a000120 :: Integral i => [i]
a000142 :: (Enum n, Num n) => [n]
a000201 :: Integral n => [n]
a000204 :: Num n => [n]
a000213 :: Num n => [n]
a000217 :: (Enum n, Num n) => [n]
a000225 :: Num n => [n]
a000285 :: Num n => [n]
a000290 :: Integral n => [n]
a000292 :: (Enum n, Num n) => [n]
a000384 :: Integral n => [n]
a000578 :: Num n => [n]
a000583 :: Integral n => [n]
a000670 :: Integral n => [n]
a000796 :: Integral n => [n]
a000930 :: Num n => [n]
a000931 :: Num n => [n]
a001008 :: Integral i => [i]
a001037 :: Integral n => [n]
a001037_n :: Integral n => n -> n
a001113 :: Integral n => [n]
a001147 :: Integral t => [t]
a001156 :: Num n => [n]
a001333 :: Num n => [n]
a001622 :: Num n => [n]
a001622_k :: Floating n => n
a001644 :: Num n => [n]
a001653 :: [Integer]
a001687 :: Num n => [n]
a001950 :: Integral n => [n]
a002267 :: Num n => [n]
a002487 :: Num n => [n]
a002858 :: [Integer]
ulam :: Int -> Integer -> [Integer] -> [Integer]
a003108 :: Num n => [n]
a003215_n :: Num n => n -> n
a003215 :: (Enum n, Num n) => [n]
a003269 :: Num n => [n]
a003520 :: Num n => [n]
a003462 :: [Integer]
a003462_n :: Integer -> Integer
a003586 :: [Integer]
a003849 :: Num n => [n]
a004001 :: [Int]
a004718 :: Num n => [n]
a005185 :: [Int]
a005448 :: Integral n => [n]
a005448_n :: Integral n => n -> n
a005728 :: Integral i => [i]
a005811 :: Integral n => [n]
a005917 :: Integral n => [n]
a006003 :: Integral n => [n]
a006003_n :: Integral n => n -> n
a006046 :: [Int]
a006052 :: Integral n => [n]
a006842 :: Integral i => [i]
a006843 :: Integral i => [i]
a007318 :: Integral i => [i]
a007318_tbl :: Integral i => [[i]]
a008277 :: (Enum n, Num n) => [n]
a008277_tbl :: (Enum n, Num n) => [[n]]
a008278 :: (Enum n, Num n) => [n]
a008278_tbl :: (Enum n, Num n) => [[n]]
a008683 :: Integral n => [n]
a008683_n :: Integral n => n -> n
a010049 :: Num n => [n]
a010060 :: [Integer]
a014081 :: (Integral i, Bits i) => [i]
a014577 :: Integral i => [i]
a016813 :: Integral n => [n]
a017817 :: Num n => [n]
a020695 :: Num n => [n]
a020985 :: [Integer]
a022095 :: Num n => [n]
a022096 :: Num n => [n]
a027750 :: Integral n => [n]
a027750_row :: Integral n => n -> [n]
a027934 :: Num n => [n]
a029635 :: Num i => [i]
a029635_tbl :: Num i => [[i]]
a030308 :: (Eq n, Num n) => [[n]]
a033622 :: [Integer]
a033622_n :: Integer -> Integer
a033812 :: Num n => [n]
a034968 :: Integral n => [n]
a036562 :: [Integer]
a036562_n :: Integer -> Integer
a046042 :: Num n => [n]
a047999 :: [Int]
a047999_tbl :: [[Int]]
a048993 :: (Enum n, Num n) => [n]
a048993_tbl :: (Enum n, Num n) => [[n]]
a049455 :: Integral n => [n]
a049456 :: Integral n => [n]
a053121 :: Num n => [n]
a053121_tbl :: Num n => [[n]]
a058265 :: Num n => [n]
a058265_k :: Floating n => n
a060588a :: Integral n => [n]
a060588a_n :: Integral n => n -> n
a061654 :: Integral n => [n]
a061654_n :: Integral n => n -> n
a071996 :: Integral n => [n]
a073334 :: Num n => [n]
a080843 :: Integral n => [n]
a080992 :: Num n => [n]
a083866 :: (Enum n, Num n) => [n]
a095660 :: Num i => [i]
a095660_tbl :: Num i => [[i]]
a095666 :: Num i => [i]
a095666_tbl :: Num i => [[i]]
a096940 :: Num i => [i]
a096940_tbl :: Num i => [[i]]
a105809 :: Num n => [n]
a105809_tbl :: Num n => [[n]]
a124010 :: Integral n => [n]
a124010_row :: Integral n => n -> [n]
a124472 :: Num n => [n]
a125519 :: Num n => [n]
a126275 :: Integral n => [n]
a126275_n :: Integral n => n -> n
a126276 :: Integral n => [n]
a126276_n :: Integral n => n -> n
a126651 :: Num n => [n]
a126652 :: Num n => [n]
a126653 :: Num n => [n]
a126654 :: Num n => [n]
a126709 :: Num n => [n]
a126710 :: Num n => [n]
a126976 :: Num n => [n]
a212804 :: Integral n => [n]
a245553 :: Integral n => [n]
a255723 :: Num n => [n]
a256184 :: Num n => [n]
a256185 :: Num n => [n]
a270876 :: Integral n => [n]
a320872 :: Num n => [n]

Documentation

a000005 :: Integral n => [n] Source #

http://oeis.org/A000005

d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. (Formerly M0246 N0086)

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 8: isPrefixOf a000005

a000010 :: Integral n => [n] Source #

http://oeis.org/A000010

Euler totient function phi(n): count numbers <= n and prime to n.

[1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12] `isPrefixOf` a000010

a000010_n :: Integral n => n -> n Source #

a000012 :: Num n => [n] Source #

http://oeis.org/A000012

The simplest sequence of positive numbers: the all 1's sequence.

a000031 :: Integral n => [n] Source #

https://oeis.org/A000031

Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

[1,2,3,4,6,8,14,20,36,60,108,188,352,632,1182,2192,4116,7712,14602,27596] `isPrefixOf` a000031

a000031_n :: Integral n => n -> n Source #

a000032 :: Num n => [n] Source #

http://oeis.org/A000032

Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1. (Formerly M0155)

[2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127] `isPrefixOf` a000032

a000040 :: Integral n => [n] Source #

http://oeis.org/A000040

The prime numbers.

[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103] `isPrefixOf` a000040

a000041 :: Num n => [n] Source #

http://oeis.org/A000041

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

1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255: isPrefixOf a000041

a000045 :: Num n => [n] Source #

http://oeis.org/A000045

Fibonacci numbers

[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946] `isPrefixOf` a000045

a000051 :: Num n => [n] Source #

http://oeis.org/A000051

a(n) = 2^n + 1

[2,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,131073] `isPrefixOf` a000051

a000071 :: Num n => [n] Source #

http://oeis.org/A000071

a(n) = Fibonacci(n) - 1.

[0,0,1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,10945,17710] `isPrefixOf` a000071

a000073 :: Num n => [n] Source #

http://oeis.org/A000073

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.

[0,0,1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,35890] `isPrefixOf` a000073

a000078 :: Num n => [n] Source #

http://oeis.org/A000078

Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0)=a(1)=a(2)=0, a(3)=1.

[0,0,0,1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536,10671,20569,39648] `isPrefixOf` a000078

a000079 :: Num n => [n] Source #

http://oeis.org/A000079

Powers of 2: a(n) = 2^n. (Formerly M1129 N0432)

[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536] `isPrefixOf` a000079
[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536] `isPrefixOf` map (2 ^) [0..]

a000085 :: Integral n => [n] Source #

http://oeis.org/A000085

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

[1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480,10349536] `isPrefixOf` a000085

a000108 :: Num n => [n] Source #

http://oeis.org/A000108

Catalan numbers: C(n) = binomial(2n,n)(n+1) = (2n)!(n!(n+1)!).

[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845] `isPrefixOf` a000108

a000120 :: Integral i => [i] Source #

http://oeis.org/A000120

1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).

[0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3] `isPrefixOf` a000120

a000142 :: (Enum n, Num n) => [n] Source #

http://oeis.org/A000142

Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).

[1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,6227020800] `isPrefixOf` a000142

a000201 :: Integral n => [n] Source #

https://oeis.org/A000201

Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = A001622

[1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,37,38,40,42] `isPrefixOf` a000201

import Sound.SC3.Plot 
plot_p1_imp [take 128 a000201 :: [Int]]

a000204 :: Num n => [n] Source #

https://oeis.org/A000204

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

[1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127] `isPrefixOf` a000204

a000213 :: Num n => [n] Source #

http://oeis.org/A000213

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

1,1,1,3,5,9,17,31,57,105,193,355,653,1201,2209,4063,7473,13745,25281,46499: isPrefixOf a000213

a000217 :: (Enum n, Num n) => [n] Source #

https://oeis.org/A000217

Triangular numbers: a(n) = binomial(n+1,2) = n(n+1)/2 = 0 + 1 + 2 + ... + n.

[0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,276] `isPrefixOf` a000217

a000225 :: Num n => [n] Source #

http://oeis.org/A000225

a(n) = 2^n - 1 (Sometimes called Mersenne numbers, although that name is usually reserved for A001348)

[0,1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535] `isPrefixOf` a000225

a000285 :: Num n => [n] Source #

http://oeis.org/000285

a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2. (Formerly M3246 N1309)

[1,4,5,9,14,23,37,60,97,157,254,411,665,1076,1741,2817,4558,7375,11933,19308] `isPrefixOf` a000285

a000290 :: Integral n => [n] Source #

http://oeis.org/A000290

The squares of the non-negative integers.

[0,1,4,9,16,25,36,49,64,81,100] `isPrefixOf` a000290

a000292 :: (Enum n, Num n) => [n] Source #

https://oeis.org/A000292

Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.

[0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,1140,1330,1540] `isPrefixOf` a000292

a000384 :: Integral n => [n] Source #

http://oeis.org/A000384

Hexagonal numbers: a(n) = n*(2*n-1). (Formerly M4108 N1705)

[0,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,780,861] `isPrefixOf` a000384

a000578 :: Num n => [n] Source #

http://oeis.org/A000578

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

[0,1,8,27,64,125,216,343,512,729,1000,1331,1728,2197,2744,3375,4096,4913,5832] `isPrefixOf` a000578

a000583 :: Integral n => [n] Source #

http://oeis.org/A000583

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

[0,1,16,81,256,625,1296,2401,4096,6561,10000,14641,20736,28561,38416,50625] `isPrefixOf` a000583

a000670 :: Integral n => [n] Source #

http://oeis.org/A000670

Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n].

[1,1,3,13,75,541,4683,47293,545835,7087261,102247563,1622632573,28091567595] `isPrefixOf` a000670

a000796 :: Integral n => [n] Source #

https://oeis.org/A000796

Decimal expansion of Pi (or digits of Pi).

[3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8,4,1,9] `isPrefixOf` a000796

pi :: Data.Number.Fixed.Fixed Data.Number.Fixed.Prec500

a000930 :: Num n => [n] Source #

https://oeis.org/A000930

Narayana's cows sequence.

[1,1,1,2,3,4,6,9,13,19,28,41,60] `isPrefixOf` a000930

a000931 :: Num n => [n] Source #

https://oeis.org/A000931

Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 0.

[1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265] `isPrefixOf` a000931

a001008 :: Integral i => [i] Source #

https://oeis.org/A001008

Numerators of harmonic numbers H(n) = Sum_{i=1..n} 1/i

1,3,11,25,137,49,363,761,7129,7381,83711,86021,1145993,1171733,1195757,2436559: isPrefixOf a001008

a001037 :: Integral n => [n] Source #

http://oeis.org/A001037

Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

[1,2,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710,14532,27594,52377,99858,190557,364722,698870] `isPrefixOf` a001037

a001037_n :: Integral n => n -> n Source #

a001113 :: Integral n => [n] Source #

http://oeis.org/A001113

Decimal expansion of e.

[2,7,1,8,2,8,1,8,2,8,4,5,9,0,4,5,2,3,5,3,6,0,2,8,7,4,7,1,3,5,2,6,6,2,4,9,7,7,5] `isPrefixOf` a001113

exp 1 :: Data.Number.Fixed.Fixed Data.Number.Fixed.Prec500

a001147 :: Integral t => [t] Source #

https://oeis.org/A001147

Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). (Formerly M3002 N1217)

[1,1,3,15,105,945,10395,135135,2027025,34459425,654729075,13749310575] `isPrefixOf` a001147

a001156 :: Num n => [n] Source #

https://oeis.org/A001156

Number of partitions of n into squares.

[1,1,1,1,2,2,2,2,3,4,4,4,5,6,6,6,8,9,10,10,12,13,14,14,16,19,20,21,23,26,27,28] `isPrefixOf` a001156

a001333 :: Num n => [n] Source #

https://oeis.org/A001333

Numerators of continued fraction convergents to sqrt(2).

1,1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,665857: isPrefixOf a001333

a001622 :: Num n => [n] Source #

http://oeis.org/A001622

Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.

[1,6,1,8,0,3,3,9,8,8,7,4,9,8,9,4,8,4,8,2,0,4,5,8,6,8,3,4,3,6,5,6,3,8,1,1,7,7,2] `isPrefixOf` a001622

a001622_k :: Data.Number.Fixed.Fixed Data.Number.Fixed.Prec500

a001622_k :: Floating n => n Source #

a001644 :: Num n => [n] Source #

http://oeis.org/A001644

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

3,1,3,7,11,21,39,71,131,241,443,815,1499,2757,5071,9327,17155,31553,58035,106743: isPrefixOf a001644

a001653 :: [Integer] Source #

https://oeis.org/A001653

Numbers k such that 2*k^2 - 1 is a square.

[1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, 38613965, 225058681, 1311738121, 7645370045, 44560482149] `isPrefixOf` a001653

a001687 :: Num n => [n] Source #

http://oeis.org/A001687

a(n) = a(n-2) + a(n-5).

0,1,0,1,0,1,1,1,2,1,3,2,4,4,5,7,7,11,11,16,18,23,29,34,45,52,68,81,102,126,154: isPrefixOf a001687

a001950 :: Integral n => [n] Source #

https://oeis.org/A001950

Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2

[2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,62,65] `isPrefixOf` a001950

a002267 :: Num n => [n] Source #

http://oeis.org/A002267

The 15 supersingular primes.

a002487 :: Num n => [n] Source #

https://oeis.org/A002487

Stern's diatomic series (or Stern-Brocot sequence)

[0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5] `isPrefixOf` a002487

a002858 :: [Integer] Source #

https://oeis.org/A002858

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

[1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126] `isPrefixOf` a002858

ulam :: Int -> Integer -> [Integer] -> [Integer] Source #

a003108 :: Num n => [n] Source #

http://oeis.org/A003108

Number of partitions of n into cubes.

[1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,7] `isPrefixOf` a003108

a003215_n :: Num n => n -> n Source #

a003215 :: (Enum n, Num n) => [n] Source #

http://oeis.org/A003215

Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).

[1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,1027,1141] `isPrefixOf` a003215

a003269 :: Num n => [n] Source #

http://oeis.org/A003269

[0,1,1,1,1,2,3,4,5,7,10,14,19,26,36,50,69,95,131,181,250,345,476,657] `isPrefixOf` a003269

a003520 :: Num n => [n] Source #

http://oeis.org/A003520

a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

[1,1,1,1,1,2,3,4,5,6,8,11,15,20,26,34,45,60,80,106,140,185,245,325,431] `isPrefixOf` a003520

a003462 :: [Integer] Source #

http://oeis.org/A003462

a(n) = (3^n - 1)/2. (Formerly M3463)

0, 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453: isPrefixOf a003462

a003462_n :: Integer -> Integer Source #

a003586 :: [Integer] Source #

http://oeis.org/A003586

3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162: isPrefixOf a003586

a003849 :: Num n => [n] Source #

https://oeis.org/A003849

The infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).

[0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0] `isPrefixOf` a003849

a004001 :: [Int] Source #

http://oeis.org/A004001

Hofstadter-Conway sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1.

[1,1,2,2,3,4,4,4,5,6,7,7,8,8,8,8,9,10,11,12,12,13,14,14,15,15,15,16,16,16,16,16] `isPrefixOf` a004001

plot_p1_ln [take 250 a004001]
plot_p1_ln [zipWith (-) a004001 (map (`div` 2) [1 .. 2000])]

a004718 :: Num n => [n] Source #

http://oeis.org/A004718

Per Nørgård's "infinity sequence"

take 32 a004718 == [0,1,-1,2,1,0,-2,3,-1,2,0,1,2,-1,-3,4,1,0,-2,3,0,1,-1,2,-2,3,1,0,3,-2,-4,5]

plot_p1_imp [take 1024 a004718]

https://www.tandfonline.com/doi/abs/10.1080/17459737.2017.1299807 https://arxiv.org/pdf/1402.3091.pdf

a005185 :: [Int] Source #

http://oeis.org/A005185

Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

[1,1,2,3,3,4,5,5,6,6,6,8,8,8,10,9,10,11,11,12,12,12,12,16,14,14,16,16,16,16,20] `isPrefixOf` a005185

a005448 :: Integral n => [n] Source #

https://oeis.org/A005448

Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

[1,4,10,19,31,46,64,85,109,136,166,199,235,274,316,361,409,460,514,571,631,694] `isPrefixOf` a005448

map a005448_n [1 .. 1000] `isPrefixOf` a005448

a005448_n :: Integral n => n -> n Source #

a005728 :: Integral i => [i] Source #

http://oeis.org/A005728

Number of fractions in Farey series of order n.

[1,2,3,5,7,11,13,19,23,29,33,43,47,59,65,73,81,97,103,121,129,141,151] `isPrefixOf` a005728

a005811 :: Integral n => [n] Source #

http://oeis.org/A005811

Number of runs in binary expansion of n (n>0); number of 1's in Gray code for n

take 32 a005811 == [0,1,2,1,2,3,2,1,2,3,4,3,2,3,2,1,2,3,4,3,4,5,4,3,2,3,4,3,2,3,2,1]

a005917 :: Integral n => [n] Source #

http://oeis.org/A005917

Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

[1,15,65,175,369,671,1105,1695,2465,3439,4641,6095,7825,9855,12209,14911,17985] `isPrefixOf` a005917

a006003 :: Integral n => [n] Source #

https://oeis.org/A006003

a(n) = n*(n^2 + 1)/2.

[0,1,5,15,34,65,111,175,260,369,505,671,870,1105,1379,1695,2056,2465,2925,3439] `isPrefixOf` a006003

map a006003_n [0 .. 1000] `isPrefixOf` a006003

a006003_n :: Integral n => n -> n Source #

a006046 :: [Int] Source #

http://oeis.org/A006046

Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1).

[0,1,3,5,9,11,15,19,27,29,33,37,45,49,57,65,81,83,87,91,99,103,111,119,135,139] `isPrefixOf` a006046

import Sound.SC3.Plot 
plot_p1_ln [take 250 a006046]
let t = log 3 / log 2
plot_p1_ln [zipWith (/) (map fromIntegral a006046) (map (\n -> n ** t) [0.0,1 .. 200])]

a006052 :: Integral n => [n] Source #

http://oeis.org/A006052

Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

[1,0,1,880,275305224] == a006052

a006842 :: Integral i => [i] Source #

http://oeis.org/A006842

Triangle read by rows: row n gives numerators of Farey series of order n.

[0,1,0,1,1,0,1,1,2,1,0,1,1,1,2,3,1,0,1,1,1,2,1,3,2,3,4,1,0,1,1,1,1,2,1,3] `isPrefixOf` a006842
plot_p1_imp [take 200 (a006842 :: [Int])]
plot_p1_pt [take 10000 (a006842 :: [Int])]

a006843 :: Integral i => [i] Source #

http://oeis.org/A006843

Triangle read by rows: row n gives denominators of Farey series of order n

[1,1,1,2,1,1,3,2,3,1,1,4,3,2,3,4,1,1,5,4,3,5,2,5,3,4,5,1,1,6,5,4,3,5,2,5] `isPrefixOf` a006843
plot_p1_imp [take 200 (a006843 :: [Int])]
plot_p1_pt [take 10000 (a006843 :: [Int])]

a007318 :: Integral i => [i] Source #

https://oeis.org/A007318

Pascal's triangle read by rows

[1: ,[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1],[1,5,10,10,5,1]] isPrefixOf a007318_tbl

a007318_tbl :: Integral i => [[i]] Source #

a008277 :: (Enum n, Num n) => [n] Source #

https://oeis.org/A008277

Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.

1,1,1,1,3,1,1,7,6,1,1,15,25,10,1,1,31,90,65,15,1,1,63,301,350,140,21,1: isPrefixOf a008277

a008277_tbl :: (Enum n, Num n) => [[n]] Source #

a008278 :: (Enum n, Num n) => [n] Source #

http://oeis.org/A008278

Triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1<=k<=n.

1,1,1,1,3,1,1,6,7,1,1,10,25,15,1,1,15,65,90,31,1,1,21,140,350,301,63,1: isPrefixOf a008278

a008278_tbl :: (Enum n, Num n) => [[n]] Source #

a008683 :: Integral n => [n] Source #

http://oeis.org/A008683

Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

[1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1,-1,0,0,1,0,0,-1,-1,-1,0,1] `isPrefixOf` a008683

a008683_n :: Integral n => n -> n Source #

a010049 :: Num n => [n] Source #

http://oeis.org/A010049

Second-order Fibonacci numbers.

[0,1,1,3,5,10,18,33,59,105,185,324,564,977,1685,2895,4957,8462,14406,24465,41455] `isInfixOf` a010049

a010060 :: [Integer] Source #

https://oeis.org/A010060

Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.

0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0: isPrefixOf a010060

a014081 :: (Integral i, Bits i) => [i] Source #

https://oeis.org/A014081

a(n) is the number of occurrences of '11' in binary expansion of n.

[0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 1, 1, 2, 2, 2, 3, 4, 0, 0, 0, 1, 0, 0, 1, 2] `isPrefixOf` a014081

a014577 :: Integral i => [i] Source #

https://oeis.org/A014577

The regular paper-folding sequence (or dragon curve sequence).

[1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1] `isPrefixOf` a014577

a016813 :: Integral n => [n] Source #

http://oeis.org/A016813

a(n) = 4*n + 1.

[1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77,81,85,89,93,97,101] `isPrefixOf` a016813

a017817 :: Num n => [n] Source #

http://oeis.org/A017817

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

[1,0,0,1,1,0,1,2,1,1,3,3,2,4,6,5,6,10,11,11,16,21,22,27,37,43,49,64,80,92] `isPrefixOf` a017817

a020695 :: Num n => [n] Source #

http://oeis.org/A020695

Pisot sequence E(2,3).

[2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711] `isPrefixOf` a020695

a020985 :: [Integer] Source #

https://oeis.org/A020985

The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials). 45

[1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1] `isPrefixOf` a020985

a022095 :: Num n => [n] Source #

http://oeis.org/A022095

Fibonacci sequence beginning 1, 5.

[1,5,6,11,17,28,45,73,118,191,309,500,809,1309,2118,3427,5545,8972,14517,23489] `isPrefixOf` a022095

a022096 :: Num n => [n] Source #

http://oeis.org/A022096

Fibonacci sequence beginning 1, 6.

[1,6,7,13,20,33,53,86,139,225,364,589,953,1542,2495,4037,6532,10569,17101,27670] `isPrefixOf` a022096

a027750 :: Integral n => [n] Source #

https://oeis.org/A027750

Triangle read by rows in which row n lists the divisors of n.

[1,1,2,1,3,1,2,4,1,5,1,2,3,6,1,7,1,2,4,8,1,3,9,1,2,5,10,1,11,1,2,3,4,6,12,1,13] `isPrefixOf` a027750

a027750_row :: Integral n => n -> [n] Source #

a027934 :: Num n => [n] Source #

http://oeis.org/A027934

a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).

[0,1,2,5,11,24,51,107,222,457,935,1904,3863,7815,15774,31781,63939,128488] `isPrefixOf` a027934

a029635 :: Num i => [i] Source #

http://oeis.org/A029635

The (1,2)-Pascal triangle (or Lucas triangle) read by rows.

[2,1,2,1,3,2,1,4,5,2,1,5,9,7,2,1,6,14,16,9,2,1,7,20,30,25,11,2,1,8,27,50,55,36] `isPrefixOf` a029635
take 7 a029635_tbl == [[2],[1,2],[1,3,2],[1,4,5,2],[1,5,9,7,2],[1,6,14,16,9,2],[1,7,20,30,25,11,2]]

a029635_tbl :: Num i => [[i]] Source #

a030308 :: (Eq n, Num n) => [[n]] Source #

http://oeis.org/A030308

Triangle T(n,k): Write n in base 2, reverse order of digits, to get the n-th row

take 9 a030308 == [[0],[1],[0,1],[1,1],[0,0,1],[1,0,1],[0,1,1],[1,1,1],[0,0,0,1]]

a033622 :: [Integer] Source #

https://oeis.org/A033622

Good sequence of increments for Shell sort (best on big values).

1, 5, 19, 41, 109, 209, 505, 929, 2161, 3905, 8929, 16001, 36289, 64769, 146305, 260609, 587521: isPrefixOf a033622

a033622_n :: Integer -> Integer Source #

a033812 :: Num n => [n] Source #

http://oeis.org/A033812

The Loh-Shu 3 X 3 magic square, lexicographically largest variant when read by columns.

a034968 :: Integral n => [n] Source #

http://oeis.org/A034968

Minimal number of factorials that add to n.

[0,1,1,2,2,3,1,2,2,3,3,4,2,3,3,4,4,5,3,4,4,5,5,6,1,2,2,3,3,4,2,3,3,4,4,5,3,4,4] `isPrefixOf` a034968

a036562 :: [Integer] Source #

https://oeis.org/A036562

a(n) = 4^(n+1) + 3*2^n + 1

1, 8, 23, 77, 281, 1073, 4193, 16577, 65921, 262913, 1050113, 4197377, 16783361, 67121153: isPrefixOf a036562

a036562_n :: Integer -> Integer Source #

a046042 :: Num n => [n] Source #

http://oeis.org/A046042

Number of partitions of n into fourth powers.

[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3] `isPrefixOf` a046042

a047999 :: [Int] Source #

http://oeis.org/A047999

Sierpiński's triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle mod 2.

[1,1,1,1,0,1,1,1,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,0,0] `isPrefixOf` a047999

a047999_tbl :: [[Int]] Source #

a048993 :: (Enum n, Num n) => [n] Source #

https://oeis.org/A048993

Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.

[1,0,1,0,1,1,0,1,3,1,0,1,7,6,1,0,1,15,25,10,1,0,1,31,90,65,15,1] `isPrefixOf` a048993

a048993_tbl :: (Enum n, Num n) => [[n]] Source #

a049455 :: Integral n => [n] Source #

http://oeis.org/A049455

Triangle read by rows, numerator of fractions of a variant of the Farey series.

[0,1,0,1,1,0,1,1,2,1,0,1,1,2,1,3,2,3,1,0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,0] `isPrefixOf` a049455
plot_p1_imp [take 200 (a049455 :: [Int])]
plot_p1_pt [take 10000 (a049455 :: [Int])]

a049456 :: Integral n => [n] Source #

http://oeis.org/A049456

Triangle read by rows, denominator of fractions of a variant of the Farey series.

1,1,1,2,1,1,3,2,3,1,1,4,3,5,2,5,3,4,1,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,1,6,5,9: isPrefixOf a049456 > plot_p1_imp [take 200 (a049456 :: [Int])] > plot_p1_pt [take 10000 (a049456 :: [Int])]

a053121 :: Num n => [n] Source #

http://oeis.org/A053121

Catalan triangle (with 0's) read by rows.

[1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,5,0,4,0,1,5,0,9,0,5,0,1,0,14,0,14,0,6,0,1,14,0] `isPrefixOf` a053121
take 7 a053121_tbl == [[1],[0,1],[1,0,1],[0,2,0,1],[2,0,3,0,1],[0,5,0,4,0,1],[5,0,9,0,5,0,1]]

a053121_tbl :: Num n => [[n]] Source #

a058265 :: Num n => [n] Source #

http://oeis.org/A058265

Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.

[1,8,3,9,2,8,6,7,5,5,2,1,4,1,6,1,1,3,2,5,5,1,8,5,2,5,6,4,6,5,3,2,8,6,6,0,0,4,2] `isPrefixOf` a058265

a058265_k :: Data.Number.Fixed.Fixed Data.Number.Fixed.Prec500

a058265_k :: Floating n => n Source #

A058265 as Floating calculation, see Data.Number.Fixed.

a060588a :: Integral n => [n] Source #

http://oeis.org/A060588

If the final two digits of n written in base 3 are the same then this digit, otherwise mod 3-sum of these two digits.

[0,2,1,2,1,0,1,0,2,0,2,1,2,1,0,1,0,2,0,2,1,2,1,0,1,0,2,0,2,1,2,1,0,1,0,2,0,2,1] `isPrefixOf` a060588a

a060588a_n :: Integral n => n -> n Source #

a061654 :: Integral n => [n] Source #

http://oeis.org/A061654

a(n) = (3*16^n + 2)/5

[1,10,154,2458,39322,629146,10066330,161061274,2576980378,41231686042] `isPrefixOf` a061654

a061654_n :: Integral n => n -> n Source #

a071996 :: Integral n => [n] Source #

http://oeis.org/A071996

a(1) = 0, a(2) = 1, a(n) = a(floor(n3)) + a(n - floor(n3)).

[0,1,1,1,1,2,2,3,3,3,4,4,4,4,4,5,5,6,6,6,6,6,7,8,8,9,9,9,9,9,9,9,10,11,12,12,12] `isPrefixOf` a071996

plot_p1_ln [take 50 a000201 :: [Int]]
plot_p1_imp [map length (take 250 (group a071996))]

a073334 :: Num n => [n] Source #

http://oeis.org/A073334

The "rhythmic infinity system" of Danish composer Per Nørgård

take 24 a073334 == [3,5,8,5,8,13,8,5,8,13,21,13,8,13,8,5,8,13,21,13,21,34,21,13]
plot_p1_imp [take 200 (a073334 :: [Int])]

a080843 :: Integral n => [n] Source #

https://oeis.org/A080843

Tribonacci word: limit S(infinity), where S(0) = 0, S(1) = 0,1, S(2) = 0,1,0,2 and for n >= 0, S(n+3) = S(n+2) S(n+1) S(n).

[0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1] `isPrefixOf` a080843

a080992 :: Num n => [n] Source #

http://oeis.org/A080992

Entries in Durer's magic square.

[16,3,2,13,5,10,11,8,9,6,7,12,4,15,14,1] == a080992

a083866 :: (Enum n, Num n) => [n] Source #

http://oeis.org/A083866

Positions of zeros in Per Nørgård's infinity sequence (A004718).

take 24 a083866 == [0,5,10,17,20,27,34,40,45,54,65,68,75,80,85,90,99,105,108,119,130,136,141,150]

a095660 :: Num i => [i] Source #

http://oeis.org/A095660

Pascal (1,3) triangle.

[3,1,3,1,4,3,1,5,7,3,1,6,12,10,3,1,7,18,22,13,3,1,8,25,40,35,16,3,1,9,33,65,75] `isPrefixOf` a095660
take 6 a095660_tbl == [[3],[1,3],[1,4,3],[1,5,7,3],[1,6,12,10,3],[1,7,18,22,13,3]]

a095660_tbl :: Num i => [[i]] Source #

a095666 :: Num i => [i] Source #

http://oeis.org/A095666

Pascal (1,4) triangle.

[4,1,4,1,5,4,1,6,9,4,1,7,15,13,4,1,8,22,28,17,4,1,9,30,50,45,21,4,1,10,39,80,95] `isPrefixOf` a095666
take 6 a095666_tbl == [[4],[1,4],[1,5,4],[1,6,9,4],[1,7,15,13,4],[1,8,22,28,17,4]]

a095666_tbl :: Num i => [[i]] Source #

a096940 :: Num i => [i] Source #

http://oeis.org/A096940

Pascal (1,5) triangle.

[5,1,5,1,6,5,1,7,11,5,1,8,18,16,5,1,9,26,34,21,5,1,10,35,60,55,26,5,1,11,45,95] `isPrefixOf` a096940
take 6 a096940_tbl == [[5],[1,5],[1,6,5],[1,7,11,5],[1,8,18,16,5],[1,9,26,34,21,5]]

a096940_tbl :: Num i => [[i]] Source #

a105809 :: Num n => [n] Source #

http://oeis.org/A105809

A Fibonacci-Pascal matrix.

[1,1,1,2,2,1,3,4,3,1,5,7,7,4,1,8,12,14,11,5,1,13,20,26,25,16,6,1,21,33,46,51,41] `isPrefixOf` a105809

a105809_tbl :: Num n => [[n]] Source #

a124010 :: Integral n => [n] Source #

http://oeis.org/A124010

Triangle in which first row is 0, n-th row (n>1) lists the (ordered) prime signature of n, that is, the exponents of distinct prime factors in factorization of n.

[0,1,1,2,1,1,1,1,3,2,1,1,1,2,1,1,1,1,1,1,4,1,1,2,1,2,1,1,1,1,1,1,3,1,2,1,1,3,2,1,1,1,1,1,1,5,1] `isPrefixOf` a124010

a124010_row :: Integral n => n -> [n] Source #

a124472 :: Num n => [n] Source #

https://oeis.org/A124472

Benjamin Franklin's 16 X 16 magic square read by rows.

[200,217,232,249,8,25,40,57,72,89,104,121,136,153,168,185,58,39,26,7,250,231] `isPrefixOf` a124472

a125519 :: Num n => [n] Source #

http://oeis.org/A125519

A 4 x 4 permutation-free magic square.

a126275 :: Integral n => [n] Source #

http://oeis.org/A126275

Moment of inertia of all magic squares of order n.

[5,60,340,1300,3885,9800,21840,44280,83325,147620,248820,402220,627445,949200] `isPrefixOf` a126275

a126275_n :: Integral n => n -> n Source #

a126276 :: Integral n => [n] Source #

http://oeis.org/A126276

Moment of inertia of all magic cubes of order n.

[18,504,5200,31500,136710,471968,1378944,3547800,8258250,17728920,35603568] `isPrefixOf` a126276

a126276_n :: Integral n => n -> n Source #

a126651 :: Num n => [n] Source #

http://oeis.org/A126651

A 7 x 7 magic square.

a126652 :: Num n => [n] Source #

http://oeis.org/A126652

A 3 X 3 magic square with magic sum 75: the Loh-Shu square A033812 multiplied by 5.

a126652 == map (* 5) a033812

a126653 :: Num n => [n] Source #

http://oeis.org/A126653

A 3 X 3 magic square with magic sum 45: the Loh-Shu square A033812 multiplied by 3.

a126653 == map (* 3) a033812

a126654 :: Num n => [n] Source #

http://oeis.org/A126654

A 3 x 3 magic square.

a126709 :: Num n => [n] Source #

http://oeis.org/A126709

The Loh-Shu 3 x 3 magic square, variant 2.

Loh-Shu magic square, attributed to the legendary Fu Xi (Fuh-Hi).

a126710 :: Num n => [n] Source #

http://oeis.org/A126710

Jaina inscription of the twelfth or thirteenth century, Khajuraho, India.

a126976 :: Num n => [n] Source #

http://oeis.org/A126976

A 6 x 6 magic square read by rows.

Agrippa (Magic Square of the Sun)

a212804 :: Integral n => [n] Source #

https://oeis.org/A212804

Expansion of (1 - x)/(1 - x - x^2).

1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946: isPrefixOf a212804

a245553 :: Integral n => [n] Source #

https://oeis.org/A245553

A Rauzy fractal sequence: trajectory of 1 under morphism 1 -> 2,3; 2 -> 3; 3 -> 1.

[1,2,3,2,3,3,1,2,3,3,1,3,1,1,2,3,2,3,3,1,3,1,1,2,3,3,1,1,2,3,1,2,3,2,3,3,1,2,3] `isPrefixOf` a245553

a255723 :: Num n => [n] Source #

http://oeis.org/A255723

Another variant of Per Nørgård's "infinity sequence"

take 24 a255723 == [0,-2,-1,2,-2,-4,1,0,-1,-3,0,1,2,0,-3,4,-2,-4,1,0,-4,-6,3,-2]
plot_p1_imp [take 400 (a255723 :: [Int])]

a256184 :: Num n => [n] Source #

http://oeis.org/A256184

First of two variations by Per Nørgård of his "infinity sequence"

take 24 a256184 == [0,-2,-1,2,-4,-3,1,-3,-2,-2,0,1,4,-6,-5,3,-5,-4,-1,-1,0,3,-5,-4]

a256185 :: Num n => [n] Source #

http://oeis.org/A256185

Second of two variations by Per Nørgård of his "infinity sequence"

take 24 a256185 == [0,-3,-2,3,-6,1,2,-5,0,-3,0,-5,6,-9,4,-1,-2,-3,-2,-1,-4,5,-8,3]

a270876 :: Integral n => [n] Source #

http://oeis.org/A270876

Number of magic tori of order n composed of the numbers from 1 to n^2.

[1,0,1,255,251449712] == a270876

a320872 :: Num n => [n] Source #

http://oeis.org/A320872

For all possible 3 X 3 magic squares made of primes, in order of increasing magic sum, list the lexicographically smallest representative of each equivalence class (modulo symmetries of the square), as a row of the 9 elements (3 rows of 3 elements each).