A281082 -id:A281082 - cp's OEIS frontend

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

Number of partitions of n into distinct centered pentagonal numbers (A005891).

			a(82) = 2 because we have [76, 6] and [51, 31].

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from G. C. Greubel)
Eric Weisstein's World of Mathematics, Centered Pentagonal Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005891, A218380, A280952, A281081, A281082, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(5 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

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

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 1, 0, 0, 1, 2

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

Number of partitions of n into distinct centered triangular numbers (A005448).

			a(46) = 2 because we have [46] and [31, 10, 4, 1].

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

Cf. A005448, A024940, A279278, A280950, A281082, A281083, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(3 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

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)).

A350204 a(n) is the smallest positive integer which can be represented as the sum of distinct centered square numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 66, 127, 151, 277, 212, 296, 325, 404, 332, 373, 440, 452, 458, 445, 464, 530, 505, 586, 572, 553, 578, 525, 637, 613, 632, 625, 626, 650, 692, 674, 638, 705, 686, 734, 710, 698, 789, 777, 745, 771, 817, 794, 746, 850, 770, 758, 847, 972, 908

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Centered Square Number

Cf. A001844, A097563, A281082, A350195, A350203.

A332006 Number of compositions (ordered partitions) of n into distinct centered square numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 1, 2, 6, 24, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 24, 1, 2, 0, 0, 0, 4, 12, 0, 0, 0, 6, 24, 0, 2, 6, 0, 0, 0, 12, 48, 0, 0, 0, 24, 121, 4, 6

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(19) = 6 because we have [13, 5, 1], [13, 1, 5], [5, 13, 1], [5, 1, 13], [1, 13, 5] and [1, 5, 13].

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

Cf. A001844, A006456, A280951, A281082, A282504, A331844, A331984, A332005.

cp's OEIS Frontend

A281083 Expansion of Product_{k>=0} (1 + x^(5*k*(k+1)/2+1)).

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A281081 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

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

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A350204 a(n) is the smallest positive integer which can be represented as the sum of distinct centered square numbers in exactly n ways, or 0 if no such integer exists.

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Author

Keywords

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A332006 Number of compositions (ordered partitions) of n into distinct centered square numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

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