cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 09 2016

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

A279279 Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))).

Original entry on oeis.org

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, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 09 2016

Keywords

Comments

Number of partitions of n into distinct hexagonal numbers (A000384).

Examples

			a(67) = 2 because we have [66, 1] and [45, 15, 6, 1].

Links

Crossrefs

Cf. A000384, A024940, A033461, A218380, A278949, A279280, A279281.

Programs

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

Formula

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

A279280 Expansion of Product_{k>=1} (1 + x^(k*(5*k-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

Views

Author

Ilya Gutkovskiy, Dec 09 2016

Keywords

Comments

Number of partitions of n into distinct heptagonal numbers (A000566).

Examples

			a(81) = 2 because we have [81] and [55, 18, 7, 1].

Links

Crossrefs

Cf. A000566, A024940, A033461, A218380, A279012, A279279, A279281.

Programs

  • Mathematica
    nmax = 120; CoefficientList[Series[Product[1 + x^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 14 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 4, 3, 3, 3, 2, 5, 4, 5, 4, 2, 3, 3, 6, 6, 5, 5, 4, 5, 7, 8, 8, 7, 6, 6, 6, 8, 9, 9, 9, 7, 8, 9, 9, 11, 10, 11, 11, 10, 12, 10, 14, 15, 14, 14, 11, 13, 13, 17, 17, 14, 15, 14, 17, 20, 19, 20, 20, 20, 21, 20, 21, 21, 25, 26, 23, 22, 21, 24, 27
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 14 2017

Keywords

Examples

			a(15) = 3 because we have [15], [12, 2, 1] and [7, 5, 2, 1].

Links

Crossrefs

Cf. A000009, A001318, A095699, A218379, A218380, A279221, A280952, A281083.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[Product[(1 + x^(k (3 k - 1)/2)) (1 + x^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A350196 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero pentagonal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 35, 92, 127, 144, 214, 237, 215, 249, 250, 319, 315, 354, 355, 366, 390, 391, 432, 431, 425, 475, 448, 478, 460, 482, 483, 510, 495, 537, 531, 525, 545, 570, 560, 594, 566, 581, 582, 606, 601, 595, 618, 603, 630, 602, 625, 652, 666, 657, 641
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 19 2021

Keywords

Links

Crossrefs

Cf. A000326, A097563, A218380, A274046, A350197, A350198, A350199.

A305355 Expansion of Product_{k>=1} (1 - x^(k*(3*k-1)/2)).

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 0, -1, 2, -1, 0, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, May 31 2018

Keywords

Links

Crossrefs

Product_{k>=1} (1 - x^(k*((m-2)*k-(m-4))/2)): A292518 (m=3), A276516 (m=4), this sequence (m=5).
Cf. A000326, A218379, A218380.

Programs

  • Maple
    seq(coeff(series(mul(1-x^(k*(3*k-1)/2),k=1..n), x,n+1),x,n),n=0..140); # Muniru A Asiru, May 31 2018

A332007 Number of compositions (ordered partitions) of n into distinct pentagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 2, 7, 2, 0, 0, 6, 26, 6, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 8, 24, 0, 0, 2, 8, 6, 0, 0, 0, 6, 26, 6, 0, 0, 0, 6, 30, 25, 2, 0, 2, 30, 122, 6, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(18) = 6 because we have [12, 5, 1], [12, 1, 5], [5, 12, 1], [5, 1, 12], [1, 12, 5] and [1, 5, 12].

Links

Crossrefs

Cf. A000326, A181324, A218379, A218380, A331843, A331844.

A296238 Expansion of Product_{k>0} (1 + x^(k*(3*k+1)/2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2017

Keywords

Crossrefs

Cf. A218380, A290942, A296237.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^(k*(3*k+1)/2))))
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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