A279012 -id:A279012 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..9828
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for sequences related to compositions

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

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

nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(k (5 k - 3)/2), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A279280 Expansion of Product_{k>=1} (1 + x^(k(5k-3)/2)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct heptagonal numbers (A000566).

			a(81) = 2 because we have [81] and [55, 18, 7, 1].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000566, A024940, A033461, A218380, A279012, A279279, A279281.

nmax = 120; CoefficientList[Series[Product[1 + x^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

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

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 10, 10, 11, 11, 11, 12, 12, 12, 14, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 19, 21, 21, 22, 24, 25, 26, 26, 26, 28, 28, 29, 31, 32, 33, 33, 33, 35, 35, 36, 39, 40, 42, 42, 43, 45, 46, 47, 50, 51, 53

Offset: 0

Ilya Gutkovskiy, Dec 04 2016

Number of partitions of n into nonzero octagonal numbers (A000567).

			a(9) = 2 because we have [8, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000567, A001156, A007294, A037444, A218379, A278949, A279012.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(3*t-2)>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(3*i-2))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (3 k - 2))), {k, 1, nmax}], {x, 0, nmax}], x]

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

A294621 Number of partitions of n into generalized heptagonal numbers (A085787).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 21, 23, 26, 29, 32, 35, 38, 41, 45, 49, 53, 59, 64, 69, 73, 80, 87, 94, 101, 109, 117, 125, 134, 145, 156, 167, 178, 190, 202, 217, 232, 249, 265, 282, 299, 318, 339, 361, 384, 408, 432, 457, 484, 514, 545, 578, 610, 646

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(8) = 4 because we have [7, 1], [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A007294, A085787, A095699, A279012, A294622, A294623.

nmax = 70; CoefficientList[Series[Product[1/((1 - x^(k (5 k - 3)/2)) (1 - x^(k (5 k + 3)/2))), {k, 1, nmax}], {x, 0, nmax}], x]

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

A332015 Number of compositions (ordered partitions) of n into distinct heptagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 1, 2, 0, 0, 6, 24, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 6, 25, 2

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(26) = 6 because we have [18, 7, 1], [18, 1, 7], [7, 18, 1], [7, 1, 18], [1, 18, 7] and [1, 7, 18].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for sequences related to compositions

Cf. A000566, A279012, A279280, A322799, A331843, A331844, A332007, A332014, A332016.

cp's OEIS Frontend

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

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A279280 Expansion of Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

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A279041 Expansion of Product_{k>=1} 1/(1 - x^(k*(3*k-2))).

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Maple

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A294621 Number of partitions of n into generalized heptagonal numbers (A085787).

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Mathematica

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A332015 Number of compositions (ordered partitions) of n into distinct heptagonal numbers.

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Author

Keywords

Examples

Links

Crossrefs

A279280 Expansion of Product_{k>=1} (1 + x^(k(5k-3)/2)).

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).