A280863 -id:A280863 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Dec 26 2018

nonn

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

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

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

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(5 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

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

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

Original entry on oeis.org

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)).

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)).

A331918 Number of compositions (ordered partitions) of n into distinct odd squares.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

			a(35) = 6 because we have [25, 9, 1], [25, 1, 9], [9, 25, 1], [9, 1, 25], [1, 25, 9] and [1, 9, 25].

Cf. A006456, A016754, A167661, A167700, A280863, A331844.

N:= 200: # for a(0)..a(N)
G:= mul(1+t*x^(i^2),i=1..floor(sqrt(N)),2):
F:= proc(n) local R, k, v;
  R:= coeff(G,x,n);
  add(k!*coeff(R,t,k),k=1..degree(R,t))
end proc:
F(0):= 1:
map(F, [$0..N]); # Robert Israel, Feb 03 2020

cp's OEIS Frontend

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

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

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A331918 Number of compositions (ordered partitions) of n into distinct odd squares.

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