A279279 -id:A279279 - cp's OEIS frontend

A279281 Expansion of Product_{k>=1} (1 + x^(k(3k-2))).

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

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

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

			a(105) = 2 because we have [96, 8, 1] and [65, 40].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000567, A024940, A033461, A218380, A279041, A279279, A279280.

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

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

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

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))).

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)).

A281084 Expansion of Product_{k>=0} (1 + x^(3k(k+1)+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

Number of partitions of n into distinct centered hexagonal numbers (A003215).

			a(98) = 2 because we have [91, 7] and [61, 37].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hex Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A003215, A279279, A280953, A281081, A281082, A281083.

nmax = 105; CoefficientList[Series[Product[1 + x^(3 k (k + 1) + 1), {k, 0, nmax}], {x, 0, nmax}], x]

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

A350197 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero hexagonal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 66, 126, 186, 277, 305, 377, 325, 416, 371, 445, 451, 511, 496, 536, 557, 607, 575, 653, 602, 630, 641, 682, 675, 668, 701, 710, 742, 746, 760, 767, 755, 800, 761, 874, 794, 845, 806, 846, 821, 861, 881, 880, 887, 867, 905, 906, 886, 911, 900

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A097563, A274046, A279279, A350196, A350198, A350199.

A332014 Number of compositions (ordered partitions) of n into distinct hexagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 1, 2, 0, 0, 6, 24, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 7, 26, 0, 0, 0, 0, 2, 8, 6, 0, 0, 0, 0, 6, 24

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(22) = 6 because we have [15, 6, 1], [15, 1, 6], [6, 15, 1], [6, 1, 15], [1, 15, 6] and [1, 6, 15].

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

Cf. A000384, A278949, A279279, A322798, A331843, A331844, A332007, A332015, A332016.

cp's OEIS Frontend

A279281 Expansion of Product_{k>=1} (1 + x^(k*(3*k-2))).

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

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

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A281084 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)+1)).

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A350197 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero hexagonal numbers in exactly n ways, or 0 if no such integer exists.

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Author

Keywords

Links

Crossrefs

A332014 Number of compositions (ordered partitions) of n into distinct hexagonal numbers.

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Author

Keywords

Examples

Links

Crossrefs

A279281 Expansion of Product_{k>=1} (1 + x^(k(3k-2))).

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

A281084 Expansion of Product_{k>=0} (1 + x^(3k(k+1)+1)).