A322801 -id:A322801 - cp's OEIS frontend

A322802 Number of compositions (ordered partitions) of n into centered hexagonal numbers (A003215).

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 36, 45, 56, 70, 88, 111, 140, 178, 226, 286, 361, 455, 573, 721, 909, 1148, 1451, 1834, 2318, 2928, 3695, 4661, 5880, 7420, 9366, 11826, 14935, 18860, 23812, 30059, 37941, 47888, 60445, 76302, 96327

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

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

Cf. A003215, A280863, A280953, A281084, A282502, A282504, A322798, A322801, A322803.

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(3*t*(t+1)+1>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-(3*i*(i+1)+1)), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

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

G.f.: 1/(1 - Sum_{k>=0} x^(3*k*(k+1)+1)).

A322803 Number of compositions (ordered partitions) of n into centered heptagonal numbers (A069099).

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, 36, 45, 55, 67, 82, 101, 125, 155, 192, 239, 297, 368, 455, 562, 694, 857, 1058, 1308, 1619, 2005, 2483, 3074, 3805, 4708, 5822, 7198, 8900, 11007, 13616, 16846, 20845, 25795, 31918, 39489

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Cf. A069099, A280863, A282502, A282504, A322799, A322801, A322802.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`((7*(t-1)*t+2)/2>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-(7*(i-1)*i+2)/2), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(7 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=0} x^(7*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

Ilya Gutkovskiy, Dec 29 2018

nonn

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

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

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

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

cp's OEIS Frontend

A322802 Number of compositions (ordered partitions) of n into centered hexagonal numbers (A003215).

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A322803 Number of compositions (ordered partitions) of n into centered heptagonal numbers (A069099).

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Maple

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

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