A025432 Number of partitions of n into 8 nonzero squares.

0, 0, 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, 2, 1, 2, 4, 1, 2, 3, 1, 2, 4, 2, 3, 4, 2, 4, 4, 1, 5, 5, 3, 5, 4, 4, 4, 5, 5, 5, 8, 4, 6, 8, 3, 6, 9, 5, 9, 8, 5, 9, 8, 5, 10, 11, 7, 10, 11, 8, 9, 10, 10, 12, 13, 9, 11, 14, 8, 11, 18, 10, 15, 16, 10, 17, 14, 10, 20, 17

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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=8 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), 8):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

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^2 > n, 0, b[n - i^2, i, t - 1]]]];
a[n_] := b[n, Sqrt[n] // Floor, 8];
Table[a[n], {n, 0, 120}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{8}],?(AllTrue[Sqrt[#],IntegerQ]&)],{n,0,100}] (* _Harvey P. Dale, Jul 20 2024 *)

Formula

a(n) = [x^n y^8] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(m) * A010052(o) * A010052(p) * A010052(n-i-j-k-l-m-o-p). - Wesley Ivan Hurt, Apr 19 2019