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 1-7 of 7 results.

A322798 Number of compositions (ordered partitions) of n into hexagonal numbers (A000384).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 22, 29, 37, 47, 60, 77, 101, 133, 174, 226, 292, 376, 486, 632, 823, 1072, 1394, 1808, 2342, 3036, 3939, 5116, 6648, 8636, 11211, 14548, 18875, 24493, 31795, 41283, 53604, 69594, 90338, 117251, 152184, 197540
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A000384, A006456, A023361, A181324, A278949, A279279, A322799, A322800.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(2*i-1)), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(k (2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(2*k-1))).

A322799 Number of compositions (ordered partitions) of n into heptagonal numbers (A000566).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 29, 37, 46, 57, 71, 89, 112, 143, 183, 233, 295, 372, 468, 588, 741, 937, 1188, 1506, 1908, 2414, 3049, 3848, 4857, 6136, 7757, 9812, 12414, 15702, 19852, 25089, 31703, 40061, 50631, 64004, 80923, 102318
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A000566, A006456, A023361, A181324, A279012, A279280, A322798, A322800.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(5*t-3)/2>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(5*i-3)/2), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(k (5 k - 3)/2), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(5*k-3)/2)).

A322800 Number of compositions (ordered partitions) of n into octagonal numbers (A000567).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 37, 46, 56, 68, 83, 102, 126, 156, 195, 244, 304, 377, 466, 575, 709, 874, 1080, 1338, 1660, 2061, 2557, 3170, 3926, 4857, 6006, 7428, 9191, 11380, 14096, 17465, 21640, 26807, 33197, 41099
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A000567, A006456, A023361, A181324, A279041, A279281, A322798, A322799.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(3*t-2)>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(3*i-2)), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(k (3 k - 2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(3*k-2))).

A308806 Expansion of 1 / Sum_{k>=0} (-x)^(k*(3*k - 1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 23, 30, 40, 52, 67, 86, 111, 145, 188, 243, 314, 406, 527, 683, 883, 1141, 1475, 1910, 2474, 3201, 4140, 5355, 6929, 8968, 11603, 15009, 19416, 25121, 32507, 42060, 54413, 70393, 91071, 117831, 152453, 197238, 255175, 330137, 427130, 552620
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 25 2019

Keywords

Crossrefs

Cf. A000041, A000326, A006950, A181324, A208061, A317665.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/Sum[(-x)^(k (3 k - 1)/2), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=0} (-x)^A000326(k).

A322801 Number of compositions (ordered partitions) of n into centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 28, 36, 46, 59, 76, 98, 128, 167, 217, 281, 363, 468, 605, 784, 1017, 1320, 1712, 2217, 2869, 3713, 4807, 6227, 8070, 10458, 13549, 17549, 22726, 29430, 38117, 49375, 63962, 82859, 107333, 139026, 180071
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A005891, A181324, A280863, A280952, A281083, A282502, A282504, A322802, A322803.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(((t+1)*5*t+2)/2>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-((i+1)*5*i+2)/2), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(5 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=0} x^(5*k*(k+1)/2+1)).

A322853 Number of compositions (ordered partitions) of n into pentagonal pyramidal numbers (A002411).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 44, 57, 74, 96, 124, 159, 205, 265, 343, 444, 574, 740, 954, 1231, 1590, 2055, 2656, 3430, 4428, 5716, 7380, 9531, 12312, 15902, 20536, 26518, 34242, 44218, 57106, 73751, 95245, 122999, 158837, 205117
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 29 2018

Keywords

Links

Crossrefs

Cf. A002411, A181324, A279221, A282582, A322340, A322801, A322854, A322855, A322856.

Programs

  • Mathematica
    nmax = 51; CoefficientList[Series[1/(1 - Sum[x^(k^2 (k + 1)/2), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k^2*(k+1)/2)).

A332007 Number of compositions (ordered partitions) of n into distinct pentagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 2, 7, 2, 0, 0, 6, 26, 6, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 8, 24, 0, 0, 2, 8, 6, 0, 0, 0, 6, 26, 6, 0, 0, 0, 6, 30, 25, 2, 0, 2, 30, 122, 6, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(18) = 6 because we have [12, 5, 1], [12, 1, 5], [5, 12, 1], [5, 1, 12], [1, 12, 5] and [1, 5, 12].

Links

Crossrefs

Cf. A000326, A181324, A218379, A218380, A331843, A331844.
Showing 1-7 of 7 results.

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