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 1-10 of 10 results.

A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 2, 6, 5, 90, 582, 4081, 45678, 378049, 3844532, 39039539, 395170118, 4589810849, 53154371025, 660113986997, 8584476248237, 113555197832758, 1572878837435750, 22259911738401660, 324143769099772448, 4869443438412466557, 74837370448784241452, 1182177603062005007658
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

Examples

			a(3) = 6 because we have [9, 0, 0], [4, 4, 1], [4, 1, 4], [1, 4, 4], [0, 9, 0] and [0, 0, 9].

Links

Crossrefs

[x^(n^b)] (Sum_{k>=0} x^(k^b))^n: A088218 (b=1), this sequence (b=2), A298671 (b=3).
Cf. A037444, A066535, A232173, A287617, A298330, A300446.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t n!*b(n^2, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 28 2018
  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]
  • PARI
    {a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ Seiichi Manyama, Oct 28 2018

Formula

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^2))^n.

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.

A307643 Number of partitions of n^3 into exactly n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 2, 6, 14, 23, 51, 108, 228, 511, 1158, 2500, 5603, 12304, 26969, 59222, 130115, 285370, 624965, 1368603, 2987117, 6517822, 14187920, 30823278, 66834822, 144671698, 312551894, 673913968, 1450292087, 3114720013, 6676277754, 14281662079
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2019

Keywords

Examples

			9^3 =
1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 5^3 + 8^3 =
1^3 + 1^3 + 2^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 6^3 =
1^3 + 1^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 7^3 =
1^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 + 5^3 + 6^3 + 6^3 =
1^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 5^3 + 5^3 + 7^3 =
2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 6^3 + 7^3,
so a(9) = 6.

Links

Crossrefs

Cf. A000578, A030272, A259792, A298671, A298672, A307644, A319435, A320841.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^3>n, 0, b(n-i^3, i, t-1))))
    end:
a:= n-> b(n^3, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 12 2019
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] + If[i^3 > n, 0, b[n - i^3, i, t - 1]]]];
a[n_] := b[n^3, n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

Formula

a(n) = A320841(n^3,n).

Extensions

More terms from Vaclav Kotesovec, Apr 20 2019

A299169 Number of ordered ways of writing n^4 as a sum of n fourth powers of nonnegative integers.

Original entry on oeis.org

1, 1, 2, 3, 4, 35, 12, 217, 8, 58473, 7930, 572891, 5556, 122985733, 5175184, 22299917655, 579379377, 743262257063, 56837361641571, 1395217574459461, 375984668290604635, 6891217627023943395, 1297848300143194333479, 26228516046396477884555, 3686440821146129098950735
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Examples

			a(6) = 12 because we have [1296, 0, 0, 0, 0, 0], [256, 256, 256, 256, 256, 16], [256, 256, 256, 256, 16, 256], [256, 256, 256, 16, 256, 256], [256, 256, 16, 256, 256, 256], [256, 16, 256, 256, 256, 256], [16, 256, 256, 256, 256, 256], [0, 1296, 0, 0, 0, 0], [0, 0, 1296, 0, 0, 0], [0, 0, 0, 1296, 0, 0], [0, 0, 0, 0, 1296, 0] and [0, 0, 0, 0, 0, 1296].

Links

Crossrefs

Cf. A000583, A046042, A259793, A298329, A298671, A298859, A298989.

Formula

a(n) = [x^(n^4)] (Sum_{k>=0} x^(k^4))^n.

Extensions

More terms from Jinyuan Wang, Dec 21 2021

A307738 Number of partitions of n^3 into at most n cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 4, 7, 18, 36, 66, 157, 329, 728, 1611, 3655, 8062, 18154, 40358, 89807, 199778, 444419, 984422, 2183461, 4827756, 10651083, 23465459, 51576034, 113092423, 247546849, 540538832, 1177836149, 2560897979, 5555722749, 12025952101, 25976048200
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2019

Keywords

Comments

Does a(n+1) / a(n) ~ 2? - David A. Corneth, Sep 27 2019

Examples

			7^3 =
1^3 + 1^3 + 5^3 + 6^3 =
1^3 + 1^3 + 3^3 + 4^3 + 5^3 + 5^3 =
1^3 + 2^3 + 3^3 + 3^3 + 4^3 + 6^3,
so a(7) = 4.

Links

Crossrefs

Cf. A000578, A025446-A025454, A030272, A105152, A259792, A298671, A298672, A307643, A307739.

Programs

  • PARI
    a(n) = {my(res = 0); res=aIterate(n^3, 1, n); res }
aIterate(s, m, q) = { if(s == 0, return(1)); if(q == 0, return(0)); sum(i = m, sqrtnint(s, 3), aIterate(s - i^3, i, q-1) ) } \\ David A. Corneth, Sep 23 2019

Extensions

a(21)-a(36) from David A. Corneth, Sep 23 2019

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)).

A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 0, 6, 6, 20, 120, 7, 1689, 6636, 36540, 64020, 963996, 2894892, 19555965, 176079995, 955611188, 6684303780, 42462792168, 292378003753, 1886275214112, 13384059605364, 87399249887334, 624073002367892, 5080120229014734, 37587589611771480
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(3) = 6 because we have [8, 1, 0], [8, 0, 1], [1, 8, 0], [1, 0, 8], [0, 8, 1] and [0, 1, 8].

Links

Crossrefs

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298937.

Programs

  • Maple
    f:= n -> coeff(add(x^(k^3),k=0..floor(n^(2/3)))^n,x,n^2):
map(f, [$0..30]); # Robert Israel, Jan 29 2018
  • Mathematica
    Table[SeriesCoefficient[Sum[x^k^3, {k, 0, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 0, 25}]

Formula

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^3))^n.

A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 9240, 34650, 1716, 48477, 551915, 6726720, 89973520, 102639744, 1824625081, 9915389400, 30143458884, 278196062760, 1995766236541, 6611689457736, 64547920386450, 236756174748626, 2315743488707806
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(7) = 7 because we have [8, 8, 8, 8, 8, 8, 1], [8, 8, 8, 8, 8, 1, 8], [8, 8, 8, 8, 1, 8, 8], [8, 8, 8, 1, 8, 8, 8], [8, 8, 1, 8, 8, 8, 8], [8, 1, 8, 8, 8, 8, 8] and [1, 8, 8, 8, 8, 8, 8].

Links

Crossrefs

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298936.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 1, 27}]]

Formula

a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^3))^n.

A303169 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^n.

Original entry on oeis.org

1, 2, 6, 30, 241, 2093, 23059, 276056, 3657901, 51751598, 792918670, 13031054778, 228632547574, 4247832219975, 83138970732860, 1710953260292025, 36844216654753387, 827664913984323748, 19363023028132371129, 470436686367280495474, 11843579175327033093769
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Comments

Number of nonnegative solutions to (x_1)^3 + (x_2)^3 + ... + (x_n)^3 <= n^3.

Links

Crossrefs

Main diagonal of A303484.
Cf. A000578, A010052, A224214, A224215, A290054, A298671, A302863.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x) Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 20}]

A319223 Number of ordered ways of writing n^3 as a sum of n squares.

Original entry on oeis.org

1, 2, 4, 32, 24, 14112, 674368, 39801344, 2454266992, 166591027058, 12820702401872, 1156778646258336, 119773060481140800, 14004241350957965408, 1791476464655904407168, 247572699435320047056384, 36696694077934168215974368, 5825316759916541565549586176, 989291135292653632945527984868
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2018

Keywords

Links

Crossrefs

Cf. A000290, A000578, A066535, A218494, A232173, A298671, A298938, A298939.

Programs

  • Mathematica
    Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, n^3}], {n, 0, 18}]
Join[{1}, Table[SquaresR[n, n^3], {n, 18}]]

Formula

a(n) = [x^(n^3)] theta_3(x)^n, where theta_3() is the Jacobi theta function.
a(n) = [x^(n^3)] (Sum_{k=-infinity..infinity} x^(k^2))^n.
Showing 1-10 of 10 results.

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