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 1-6 of 6 results.

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)).

A322800 Number of compositions (ordered partitions) of n into octagonal numbers (A000567).

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, 37, 46, 56, 68, 83, 102, 126, 156, 195, 244, 304, 377, 466, 575, 709, 874, 1080, 1338, 1660, 2061, 2557, 3170, 3926, 4857, 6006, 7428, 9191, 11380, 14096, 17465, 21640, 26807, 33197, 41099
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A000567, A006456, A023361, A181324, A279041, A279281, A322798, A322799.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(3*t-2)>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(3*i-2)), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(k (3 k - 2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

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

A350199 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero octagonal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 105, 238, 302, 342, 406, 570, 622, 639, 686, 758, 750, 822, 815, 862, 883, 926, 983, 1027, 966, 995, 987, 1099, 1091, 1159, 1142, 1143, 1183, 1227, 1171, 1231, 1163, 1315, 1295, 1296, 1275, 1372, 1267, 1371, 1343, 1375, 1399, 1476, 1411, 1403, 1407, 1463, 1451, 1508, 1507
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 19 2021

Keywords

Links

Crossrefs

Cf. A000567, A097563, A274046, A279281, A350196, A350197, A350198.

A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 2, 3, 1, 1, 2, 2, 1, 0, 1, 3, 2, 2, 3, 1, 1, 1, 3, 5, 2, 2, 3, 2, 2, 1, 3, 5, 2, 1, 3, 3, 2, 1, 3, 6, 3, 1, 3, 4, 3, 1, 4, 7, 3, 0, 3, 6, 4, 1, 2, 7, 5, 2, 4, 5, 5, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2017

Keywords

Examples

			a(21) = 2 because we have [21] and [16, 5].

Links

Crossrefs

Cf. A001082, A024940, A279281, A290942, A294622, A294623.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(k (3 k - 2))) (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)))*(1 + x^(k*(3*k+2))).

A332016 Number of compositions (ordered partitions) of n into distinct octagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 2, 0, 0, 6, 24, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(30) = 6 because we have [21, 8, 1], [21, 1, 8], [8, 21, 1], [8, 1, 21], [1, 21, 8] and [1, 8, 21].

Links

Crossrefs

Cf. A000567, A279041, A279281, A322800, A331843, A331844, A332007, A332014, A332015.
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.