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A025430 Number of partitions of n into 6 nonzero squares.

%I A025430 #27 Jan 01 2025 03:28:20
%S A025430 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,
%T A025430 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,
%U A025430 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
%N A025430 Number of partitions of n into 6 nonzero squares.
%H A025430 Alois P. Heinz, <a href="/A025430/b025430.txt">Table of n, a(n) for n = 0..10000</a>
%H A025430 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025430 a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 19 2019
%F A025430 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
%p A025430 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A025430       `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A025430       `if`(i^2>n, 0, b(n-i^2, i, t-1))))
%p A025430     end:
%p A025430 a:= n-> b(n, isqrt(n), 6):
%p A025430 seq(a(n), n=0..120);  # _Alois P. Heinz_, May 30 2014
%t A025430 a[n_] := Count[PowersRepresentations[n, 6, 2], r_ /; FreeQ[r, 0]]; Array[a, 120, 0] (* _Jean-François Alcover_, Feb 19 2016 *)
%t A025430 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]]]];
%t A025430 a[n_] := b[n, Sqrt[n] // Floor, 6];
%t A025430 a /@ Range[0, 120] (* _Jean-François Alcover_, Nov 06 2020, after _Alois P. Heinz_ *)
%Y A025430 Column k=6 of A243148.
%K A025430 nonn,easy
%O A025430 0,22
%A A025430 _David W. Wilson_