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 1-5 of 5 results.

A298672 Number of ordered ways of writing n^3 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 20, 0, 1121, 72828, 872640, 9037710, 118590450, 1743739426, 24407782672, 424735169040, 7802802463460, 135385454550288, 2823521345232834, 59332856029292241, 1238888844244575904, 28893281420537822022, 684650546073054870188, 16342742577592266281996
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(6) = 20 because we have [64, 64, 64, 8, 8, 8], [64, 64, 8, 64, 8, 8], [64, 64, 8, 8, 64, 8], [64, 64, 8, 8, 8, 64], [64, 8, 64, 64, 8, 8], [64, 8, 64, 8, 64, 8], [64, 8, 64, 8, 8, 64], [64, 8, 8, 64, 64, 8], [64, 8, 8, 64, 8, 64], [64, 8, 8, 8, 64, 64], [8, 64, 64, 64, 8, 8], [8, 64, 64, 8, 64, 8], [8, 64, 64, 8, 8, 64], [8, 64, 8, 64, 64, 8], [8, 64, 8, 64, 8, 64], [8, 64, 8, 8, 64, 64], [8, 8, 64, 64, 64, 8], [8, 8, 64, 64, 8, 64], [8, 8, 64, 8, 64, 64] and [8, 8, 8, 64, 64, 64].

Links

Crossrefs

Cf. A000578, A030272, A232173, A259792, A290247, A291700, A298329, A298330, A298641, A298671.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, n}]^n, {x, 0, n^3}], {n, 1, 23}]]

Formula

a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^3))^n.

A298671 Number of ordered ways of writing n^3 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 146, 4207, 26329, 257721, 3556495, 42685181, 631230381, 9409600499, 142557084957, 2781352245050, 52598395446786, 950288577530017, 20768368026768594, 448759012546543804, 9652848877533217174, 235179507693424886403, 5756272592837812726164, 140920987987840184113287
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(3) = 3 because we have [27, 0, 0], [0, 27, 0] and [0, 0, 27].

Links

Crossrefs

Cf. A000578, A030272, A232173, A259792, A290054, A290247, A291700, A298329, A298330, A298641, A298672.

Programs

  • Mathematica
    Table[SeriesCoefficient[Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 23}]

Formula

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^3))^n.

A298848 Number of partitions of n^3 into distinct cubes > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 5, 4, 3, 4, 13, 11, 15, 20, 23, 34, 52, 49, 97, 118, 164, 192, 296, 330, 525, 745, 825, 1354, 1820, 1994, 3356, 4605, 5543, 8335, 12319, 13124, 21133, 28634, 33209, 51272, 71154, 85329, 126806, 174704, 210157, 310269, 433490, 511199
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 27 2018

Keywords

Examples

			a(6) = 2 because we have [216] and [125, 64, 27].

Links

Crossrefs

Cf. A000578, A003108, A030272, A078128, A259792, A279329, A280130, A298641, A298671, A298672.

Formula

a(n) = [x^(n^3)] Product_{k>=2} (1 + x^(k^3)).
a(n) = A280130(A000578(n)).

A298989 Number of partitions of n^4 into fourth powers > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 8, 32, 101, 687, 3584, 23564, 146424, 937953, 6006835, 38521889, 247868209, 1591813628, 10234693956, 65662254277, 420757890998, 2688786485779, 17134894394402, 108819902923649, 688544716659489, 4339161392334630, 27229261402800035, 170114849290565556
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2018

Keywords

Examples

			a(4) = 2 because we have [256] and [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16].

Links

Crossrefs

Cf. A000583, A046042, A092362, A259793, A298641, A298859.

Formula

a(n) = [x^(n^4)] Product_{k>=2} 1/(1 - x^(k^4)).

Extensions

a(21)-a(27) from Alois P. Heinz, Apr 18 2019

A331899 Number of compositions (ordered partitions) of n^3 into distinct cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 127, 1, 1, 127, 769, 10945, 15961, 86641, 86521, 430717, 4140367, 4146751, 93669001, 1538834041, 663998665, 6883029151, 1014140647, 20591858857, 121532206567, 1637261351983, 2981530899847, 5950338797191, 47072230385425
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2020

Keywords

Examples

			a(6) = 7 because we have [216], [125, 64, 27], [125, 27, 64], [64, 125, 27], [64, 27, 125], [27, 125, 64] and [27, 64, 125].

Links

Crossrefs

Cf. A000578, A030272, A259792, A290247, A298641, A298672, A298848, A331845, A331884.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^3, n, 0):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 31 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n - i^3, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^3, n, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

Formula

a(n) = A331845(A000578(n)).
Showing 1-5 of 5 results.

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