%I #29 Jul 20 2024 16:32:38

%S 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,

%T 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,

%U 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

%N Number of partitions of n into 8 nonzero squares.

%H Alois P. Heinz, <a href="/A025432/b025432.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

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

%F 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

%p b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),

%p `if`(i<1 or t<1, 0, b(n, i-1, t)+

%p `if`(i^2>n, 0, b(n-i^2, i, t-1))))

%p end:

%p a:= n-> b(n, isqrt(n), 8):

%p seq(a(n), n=0..120); # _Alois P. Heinz_, May 30 2014

%t 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 a[n_] := b[n, Sqrt[n] // Floor, 8];

%t Table[a[n], {n, 0, 120}] (* _Jean-François Alcover_, May 20 2018, after _Alois P. Heinz_ *)

%t Table[Count[IntegerPartitions[n,{8}],_?(AllTrue[Sqrt[#],IntegerQ]&)],{n,0,100}] (* _Harvey P. Dale_, Jul 20 2024 *)

%Y Column k=8 of A243148.

%K nonn

%O 0,24

%A _David W. Wilson_