A025430 Number of partitions of n into 6 nonzero squares.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 4, 1, 3, 3, 1, 4, 2, 1, 4, 4, 3, 3, 3, 3, 3, 3, 3, 4, 6, 3, 4, 6, 2, 3, 7, 3, 6, 5, 2, 7, 5, 3, 7, 7, 5, 6, 6, 5, 5, 6, 6, 7, 9, 5, 6, 10, 4, 6, 11, 5, 10, 8, 6, 11, 7, 5, 11, 10, 8, 10, 8, 8, 8, 9, 10, 11, 13

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Column k=6 of A243148.

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^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 6):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

Mathematica

a[n_] := Count[PowersRepresentations[n, 6, 2], r_ /; FreeQ[r, 0]]; Array[a, 120, 0] (* Jean-François Alcover, Feb 19 2016 *)
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^2 > n, 0, b[n - i^2, i, t - 1]]]];
a[n_] := b[n, Sqrt[n] // Floor, 6];
a /@ Range[0, 120] (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(m) * A010052(n-i-j-k-l-m). - Wesley Ivan Hurt, Apr 19 2019