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.

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A035684 Number of partitions of n into parts 8k+1 and 8k+7 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4, 4, 5, 7, 10, 11, 11, 11, 11, 12, 14, 18, 23, 25, 26, 26, 27, 29, 33, 40, 47, 52, 54, 56, 58, 62, 70, 81, 93, 101, 107, 111, 116, 124, 137, 155, 172, 188, 199, 208, 218, 233, 255, 282, 311, 336, 357, 374, 393
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035683, A035685-A035699.

Programs

  • Mathematica
    nmax = 68; s1 = Range[0, nmax/8]*8 + 1; 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 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 + 7)), {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 + 7))). - Robert Price, Aug 15 2020

A035685 Number of partitions of n into parts 8k+2 and 8k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 10, 7, 12, 11, 14, 13, 18, 15, 24, 19, 28, 27, 33, 31, 42, 36, 51, 45, 60, 58, 71, 68, 87, 79, 103, 96, 120, 118, 141, 137, 169, 159, 197, 189, 228, 226, 266, 262, 314, 302, 362, 355, 416, 416, 482, 478, 561, 550
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035684, A035686-A035699.

Programs

  • Mathematica
    nmax = 68; s1 = Range[0, nmax/8]*8 + 2; 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 15 2020 *)
nmax = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 3))). - Robert Price, Aug 15 2020

A035686 Number of partitions of n into parts 8k+2 and 8k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 5, 0, 5, 0, 8, 0, 8, 0, 14, 0, 15, 0, 22, 0, 23, 0, 34, 0, 37, 0, 51, 0, 54, 0, 74, 0, 81, 0, 107, 0, 116, 0, 150, 0, 165, 0, 210, 0, 229, 0, 287, 0, 316, 0, 392, 0, 430, 0, 526, 0, 580, 0, 704, 0, 774, 0, 929, 0, 1024, 0, 1223, 0, 1347, 0
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035685, A035687-A035699.

Programs

  • Mathematica
    nmax = 77; s1 = Range[0, nmax/8]*8 + 2; s2 = Range[0, nmax/8]*8 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 4))). - Robert Price, Aug 15 2020

A035687 Number of partitions of n into parts 8k+2 and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 4, 1, 4, 3, 4, 4, 7, 4, 10, 4, 11, 8, 11, 11, 15, 12, 21, 12, 25, 18, 26, 24, 31, 28, 42, 29, 50, 38, 55, 50, 62, 58, 79, 63, 95, 76, 105, 96, 118, 113, 144, 123, 172, 145, 193, 178, 213, 208, 255, 230, 302, 262, 340, 316
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035686, A035688-A035699.

Programs

  • Mathematica
    nmax = 70; s1 = Range[0, nmax/8]*8 + 2; 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 15 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 15 2020

A035690 Number of partitions of n into parts 8k+3 and 8k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 4, 1, 3, 4, 7, 3, 4, 8, 10, 4, 8, 11, 15, 8, 11, 18, 21, 11, 19, 24, 30, 19, 25, 37, 42, 25, 40, 50, 56, 41, 53, 70, 79, 54, 77, 95, 103, 80, 103, 129, 141, 106, 144, 172, 183, 151, 189, 228, 246, 197, 257, 301, 314
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035689, A035691-A035699.

Programs

  • Mathematica
    nmax = 71; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 4))). - Robert Price, Aug 15 2020

A035691 Number of partitions of n into parts 8k+3 and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 3, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 7, 4, 8, 7, 6, 13, 8, 10, 15, 10, 17, 17, 14, 24, 19, 22, 30, 23, 33, 34, 31, 46, 39, 44, 56, 47, 63, 65, 61, 82, 75, 84, 101, 90, 113, 118, 115, 145, 137, 151, 176, 165, 197, 207, 206, 246, 242, 264
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035690, A035692-A035699.

Programs

  • Mathematica
    nmax = 74; s1 = Range[0, nmax/8]*8 + 3; 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 15 2020 *)
nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 15 2020

A035694 Number of partitions of n into parts 8k+4 and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 3, 3, 1, 1, 6, 3, 3, 1, 8, 7, 3, 3, 12, 9, 7, 3, 16, 15, 9, 7, 22, 19, 16, 9, 30, 29, 20, 16, 40, 38, 32, 20, 54, 54, 41, 33, 69, 70, 61, 42, 93, 95, 78, 64, 118, 124, 110, 81, 157, 163, 141, 117, 196, 211, 192, 149, 258
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035693, A035695-A035699.

Programs

  • Mathematica
    nmax = 77; s1 = Range[0, nmax/8]*8 + 4; 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 16 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 16 2020

A035695 Number of partitions of n into parts 8k+4 and 8k+6 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 4, 0, 3, 0, 7, 0, 4, 0, 10, 0, 8, 0, 15, 0, 11, 0, 21, 0, 18, 0, 30, 0, 24, 0, 42, 0, 37, 0, 56, 0, 50, 0, 78, 0, 70, 0, 102, 0, 95, 0, 137, 0, 129, 0, 179, 0, 171, 0, 236, 0, 227, 0, 303, 0, 297, 0, 395, 0, 386, 0, 502, 0
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Bisections give A035626 (even part), A000004 (odd part).
Cf. A035441-A035468, A035618-A035694, A035696-A035699.

Programs

  • Mathematica
    nmax = 83; s1 = Range[0, nmax/8]*8 + 4; 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 16 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 6))). - Robert Price, Aug 16 2020

A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 4, 0, 10, 0, 11, 0, 22, 0, 25, 0, 44, 0, 51, 0, 83, 0, 98, 0, 149, 0, 177, 0, 259, 0, 309, 0, 436, 0, 521, 0, 716, 0, 857, 0, 1151, 0, 1376, 0, 1816, 0, 2170, 0, 2818, 0, 3361, 0, 4309, 0, 5132, 0, 6502, 0, 7728, 0, 9695, 0, 11501, 0, 14298
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Bisections give: A006477 (even part), A000004 (odd part).
Cf. A035441-A035468, A035618-A035621, A035623-A035699.

Programs

  • Mathematica
    nmax = 70; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 0, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, 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>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 2))). - Robert Price, Aug 16 2020

A035624 Number of partitions of n into parts 4k+1 and 4k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 8, 14, 15, 22, 23, 34, 37, 51, 54, 74, 81, 107, 116, 150, 165, 210, 229, 287, 316, 392, 430, 526, 580, 704, 774, 929, 1024, 1223, 1347, 1593, 1756, 2068, 2278, 2663, 2933, 3416, 3762, 4355, 4793, 5529, 6084, 6985, 7680, 8789
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035623, A035625-A035699.

Programs

  • Mathematica
    nmax = 53; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 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 = 53; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, 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>=0} (1 - x^(4 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 2))). - Robert Price, Aug 16 2020
Previous Showing 11-20 of 81 results. Next

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