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A248801 Number of sets of nonzero squares with sum <= n.

%I A248801 #19 Jul 23 2025 11:46:50
%S A248801 1,2,2,2,3,4,4,4,4,5,6,6,6,7,8,8,9,10,10,10,11,12,12,12,12,14,16,16,
%T A248801 16,18,20,20,20,20,21,22,23,24,25,26,27,29,30,30,30,32,34,34,34,36,39,
%U A248801 40,41,43,45,46,47,48,49,50,50,52,55,56,57,61,64,64,65
%N A248801 Number of sets of nonzero squares with sum <= n.
%C A248801 Partial sums of A033461.
%H A248801 Robert Israel, <a href="/A248801/b248801.txt">Table of n, a(n) for n = 0..10000</a>
%F A248801 G.f.: (1-x)^(-1) * product(k>=1, 1 + x^(k^2)).
%e A248801 For n=5 the sets are {}, {1^2}, {2^2}, {1^2, 2^2} so a(5) = 4.
%p A248801 N:= 200: # to get a(0) to a(N)
%p A248801 g:= (1-x)^(-1)*mul(1 + x^(m^2), m=1 .. floor(sqrt(N))):
%p A248801 S:= series(g, x, N+1):
%p A248801 seq(coeff(S,x,j),j=0..N);
%t A248801 CoefficientList[Series[(1 - x)^(-1) Product[1 + x^(k^2), {k, 50}], {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 15 2014 *)
%t A248801 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-1]]]]; Table[b[n, Floor[Sqrt[n]]], {n, 0, 100}] // Accumulate (* _Jean-François Alcover_, Apr 17 2019, after _Alois P. Heinz_ in A033461 *)
%Y A248801 Cf. A033461.
%K A248801 nonn
%O A248801 0,2
%A A248801 _Robert Israel_, Oct 14 2014