cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 81 results. Next

A035699 Number of partitions of n into parts 8k+6 and 8k+7 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 3, 2, 3, 0, 1, 1, 3, 3, 6, 4, 5, 1, 3, 3, 7, 7, 11, 7, 8, 3, 7, 8, 15, 13, 19, 12, 13, 8, 16, 17, 27, 24, 30, 20, 23, 18, 32, 32, 46, 40, 48, 34, 41, 37, 56, 57, 76, 66, 76, 58, 71, 67, 97, 96, 122, 105, 119
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035698.

Programs

  • Mathematica
    nmax = 83; s1 = Range[0, nmax/8]*8 + 6; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 6)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - Robert Price, Aug 16 2020

A035619 Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 3, 1, 3, 7, 3, 8, 14, 8, 17, 26, 18, 33, 47, 36, 61, 81, 68, 106, 137, 121, 181, 224, 209, 296, 362, 347, 478, 570, 565, 750, 890, 894, 1166, 1360, 1396, 1774, 2062, 2134, 2677, 3076, 3228, 3973, 4555, 4804, 5854, 6657, 7085, 8513
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618, A035620-A035699.

Programs

  • Mathematica
    nmax = 55; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 55; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(3 k)))*(-1 + 1/Product_{k>=0} (1 - x^(3 k + 2))). - Robert Price, Aug 16 2020

A035620 Number of partitions of n into parts 3k+1 and 3k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 26, 33, 43, 55, 70, 88, 111, 137, 170, 208, 256, 311, 378, 456, 551, 658, 790, 940, 1119, 1325, 1570, 1847, 2179, 2554, 2996, 3499, 4088, 4753, 5533, 6414, 7436, 8593, 9931, 11439, 13180, 15140, 17391, 19926, 22827
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035619, A035621-A035699.

Programs

  • Mathematica
    nmax = 50; s1 = Range[0, nmax/3]*3 + 1; s2 = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 50; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(3 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(3 k + 2))). - Robert Price, Aug 16 2020

A035621 Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 10, 11, 11, 11, 22, 25, 26, 26, 44, 51, 54, 55, 84, 98, 105, 108, 153, 178, 193, 200, 269, 313, 341, 356, 459, 531, 582, 611, 764, 880, 967, 1021, 1244, 1424, 1568, 1662, 1988, 2264, 2494, 2653, 3122, 3536, 3896, 4155
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035620, A035622-A035699.

Programs

  • Mathematica
    nmax = 56; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 56; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 1))). - Robert Price, Aug 16 2020

A035623 Number of partitions of n into parts 4k and 4k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 1, 3, 6, 1, 3, 7, 12, 3, 7, 15, 21, 7, 16, 28, 36, 16, 31, 50, 60, 32, 57, 85, 98, 60, 100, 141, 157, 107, 169, 226, 248, 184, 276, 358, 385, 305, 442, 553, 591, 495, 691, 845, 896, 782, 1063, 1270, 1343, 1216, 1608, 1890, 1993
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035622, A035624-A035699.

Programs

  • Maple
    N:= 100:
P:= (-1 + 1/mul(1-x^(4*k+3), k=0..(N-3)/4))*(-1 + 1/mul(1-x^(4*k), k=1..N/4)):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Feb 23 2016
  • Mathematica
    nmax = 63; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 06 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1-x^(4k+3)))*(-1 + 1/Product_{k>=1} (1-x^(4k))). - Robert Israel, Feb 23 2016
a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(3/4) / (2^(5/4) * 3^(5/8) * Gamma(1/4) * n^(9/8)). - Vaclav Kotesovec, May 26 2018

A035626 Number of partitions of n into parts 4k+2 and 4k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 4, 3, 7, 4, 10, 8, 15, 11, 21, 18, 30, 24, 42, 37, 56, 50, 78, 70, 102, 95, 137, 129, 179, 171, 236, 227, 303, 297, 395, 386, 502, 501, 643, 641, 814, 820, 1030, 1041, 1291, 1317, 1622, 1652, 2018, 2075, 2509, 2582, 3107, 3212, 3834
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Bisection of A035695 (even part).
Cf. A035441-A035468, A035618-A035625, A035627-A035699.

Programs

  • Mathematica
    nmax = 59; s1 = Range[0, nmax/4]*4 + 2; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(4 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 3))). - Robert Price, Aug 16 2020

A035672 Number of partitions of n into parts 8k and 8k+1 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 10, 11, 11, 11, 11, 11, 11, 11, 22, 25, 26, 26, 26, 26, 26, 26, 44, 51, 54, 55, 55, 55, 55, 55, 84, 98, 105, 108, 109, 109, 109, 109, 153, 178, 193, 200, 203, 204, 204, 204, 270, 313, 341, 356, 363, 366
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035671, A035673-A035699.

Programs

  • Mathematica
    nmax = 70; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 1)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020

A035674 Number of partitions of n into parts 8k and 8k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 6, 3, 1, 7, 3, 1, 7, 3, 12, 7, 3, 15, 7, 3, 16, 7, 21, 16, 7, 28, 16, 7, 31, 16, 36, 32, 16, 50, 32, 16, 57, 32, 60, 60, 32, 85, 61, 32, 100, 61, 98, 107, 61, 141, 110, 61, 169, 111, 157, 184, 111, 226
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035673, A035675-A035699.

Programs

  • Mathematica
    nmax = 78; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020

A035676 Number of partitions of n into parts 8k and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 3, 0, 1, 0, 0, 3, 0, 1, 6, 0, 3, 0, 1, 7, 0, 3, 11, 1, 7, 0, 3, 14, 1, 7, 18, 3, 15, 1, 7, 25, 3, 15, 30, 7, 28, 3, 15, 44, 7, 29, 47, 15, 51, 7, 29, 72, 15, 54, 73, 29, 87, 15, 55, 116, 29, 94, 111, 55, 144, 29, 97, 180, 55
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035675, A035677-A035699.

Programs

  • Mathematica
    nmax = 83; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 5)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

A035683 Number of partitions of n into parts 8k+1 and 8k+6 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 4, 4, 5, 5, 7, 7, 10, 11, 12, 12, 14, 14, 18, 19, 24, 26, 29, 29, 33, 34, 41, 43, 51, 55, 61, 63, 71, 73, 85, 90, 102, 110, 122, 126, 141, 146, 164, 174, 194, 207, 230, 239, 263, 275, 304, 322, 355, 377, 414, 433, 473, 495
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035682, A035684-A035699.

Programs

  • Mathematica
    nmax = 68; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2], x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8*k + 6))). - Robert Price, Aug 15 2020
Showing 1-10 of 81 results. Next

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