A322800 -id:A322800 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..8828
Eric Weisstein's World of Mathematics, Hexagonal Number
Index entries for sequences related to compositions

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

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

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(k (2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

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

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)).

A322856 Number of compositions (ordered partitions) of n into octagonal pyramidal numbers (A002414).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64, 76, 91, 110, 135, 166, 204, 250, 305, 370, 447, 539, 650, 787, 956, 1164, 1419, 1730, 2107, 2562, 3110, 3770, 4569, 5540, 6723, 8166, 9926, 12070, 14677, 17841, 21675

Offset: 0

Ilya Gutkovskiy, Dec 29 2018

nonn

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

Cf. A002414, A279224, A280863, A282582, A322340, A322800, A322853, A322854, A322855.

nmax = 56; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A332016 Number of compositions (ordered partitions) of n into distinct octagonal numbers.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(30) = 6 because we have [21, 8, 1], [21, 1, 8], [8, 21, 1], [8, 1, 21], [1, 21, 8] and [1, 8, 21].

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

Cf. A000567, A279041, A279281, A322800, A331843, A331844, A332007, A332014, A332015.

A363275 Expansion of 1 / Sum_{k>=0} x^(k(3k - 2)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -7, 6, -4, 1, 3, -9, 16, -24, 32, -39, 44, -46, 44, -35, 18, 8, -43, 86, -135, 187, -238, 280, -304, 300, -259, 171, -28, -174, 435, -746, 1088, -1431, 1736, -1952, 2017, -1864, 1425, -641, -527, 2086, -4002

Offset: 0

Ilya Gutkovskiy, May 25 2023

sign

Cf. A000567, A106507, A195848, A308806, A317665, A322800, A361979, A363149.

nmax = 56; CoefficientList[Series[1/Sum[x^(k (3 k - 2)), {k, 0, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

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

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

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A322856 Number of compositions (ordered partitions) of n into octagonal pyramidal numbers (A002414).

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A332016 Number of compositions (ordered partitions) of n into distinct octagonal numbers.

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A363275 Expansion of 1 / Sum_{k>=0} x^(k*(3*k - 2)).

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A363275 Expansion of 1 / Sum_{k>=0} x^(k(3k - 2)).