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.

A298641 Number of partitions of n^3 into cubes > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 6, 45, 100, 377, 1181, 4063, 13225, 45218, 150928, 511970, 1717140, 5777895, 19308880, 64360153, 213446697, 705095144, 2317573307, 7583418322, 24690176885, 80003762726, 257959340058, 827713115396, 2642967441892, 8398644246488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(4) = 2 because we have [64] and [8, 8, 8, 8, 8, 8, 8, 8].

Links

Crossrefs

Cf. A000578, A003108, A030272, A078128, A092362, A259792, A279329, A280130, A290247.

Programs

  • Maple
    g:= proc(n, L) # number of partitions of n into cubes > 1 and <= L
   option remember;
   local t,k;
   t:= 0;
   if n = 0 then return 1 fi;
   if n < 8 then return 0 fi;
   for k from 2 while k^3 <= min(n,L) do
     t:= t + procname(n-k^3, k^3)
   od
end proc:
f:= n -> g(n^3, n^3):
map(f, [$0..50]); # Robert Israel, Jan 24 2018
  • Mathematica
    mx = 30; s = Series[Product[1/(1 - x^(k^3)), {k, 2, mx}], {x, 0, mx^3}]; Table[ CoefficientList[s, x][[1 + n^3]], {n, 0, mx}] (* Robert G. Wilson v, Jan 24 2018 *)

Formula

a(n) = [x^(n^3)] Product_{k>=2} 1/(1 - x^(k^3)).
a(n) = A078128(A000578(n)).
a(n) ~ exp(4*(Gamma(1/3) * Zeta(4/3))^(3/4) * n^(3/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/2) / (8 * 3^(5/2) * Pi^2 * n^6). - Vaclav Kotesovec, Jan 31 2018

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2004

Keywords

Examples

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Links

Crossrefs

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Formula

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

Extensions

More terms from Vladeta Jovovic, Mar 24 2004
Corrected a(0) by Alois P. Heinz, Apr 15 2013

A093116 Number of partitions of n^2 into squares not less than n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2004

Keywords

Examples

			n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Links

Crossrefs

Cf. A093115, A092362, A001156, A037444, A078134.
Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r+`if`(r^2Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A298642 Number of partitions of n^2 into distinct squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 5, 2, 10, 4, 12, 12, 11, 19, 23, 43, 50, 55, 78, 120, 126, 234, 207, 407, 385, 701, 712, 1090, 1231, 1850, 2102, 3054, 3385, 4988, 5584, 7985, 9746, 12205, 15737, 18968, 25157, 30927, 39043, 47708, 61915, 74592, 99554
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(5) = 2 because we have [25] and [16, 9].

Links

Crossrefs

Cf. A000290, A001156, A030273, A033461, A037444, A078134, A092362, A093115, A093116, A280129, A298640.

Formula

a(n) = [x^(n^2)] Product_{k>=2} (1 + x^(k^2)).
a(n) = A280129(A000290(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

A294071 Number of ordered ways of writing n^2 as a sum of n squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 07 2018

Keywords

Examples

			a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Links

Crossrefs

Cf. A000290, A037444, A066535, A078134, A092362, A232173, A280129, A280542, A281154, A281155, A298329, A298330, A298640, A298642.

Programs

  • Mathematica
    Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]

Formula

a(n) = [x^(n^2)] (Sum_{k>=2} x^(k^2))^n.
Showing 1-6 of 6 results.

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