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A063691 Number of solutions to x^2 + y^2 + z^2 = n in positive integers.

%I A063691 #35 Oct 23 2024 08:39:32
%S A063691 0,0,0,1,0,0,3,0,0,3,0,3,1,0,6,0,0,3,3,3,0,6,3,0,3,0,6,4,0,6,6,0,0,6,
%T A063691 3,6,3,0,9,0,0,9,6,3,3,6,6,0,1,6,6,6,0,6,12,0,6,6,0,9,0,6,12,0,0,6,12,
%U A063691 3,3,12,6,0,3,3,12,7,3,12,6,0,0,12,3,9,6,0,15,0,3,15
%N A063691 Number of solutions to x^2 + y^2 + z^2 = n in positive integers.
%H A063691 T. D. Noe, <a href="/A063691/b063691.txt">Table of n, a(n) for n = 0..10000</a>
%F A063691 G.f.: (Sum_{m>=1} x^(m^2))^3.
%e A063691 a(5)=0;
%e A063691 a(6)=3 because 1^2+1^2+2^2 = 1^2+2^2+1^2 = 2^2+1^2+1^2 = 6;
%e A063691 a(27)=4 because 1^2+1^2+5^2 = 1^2+5^2+1^2 = 3^2+3^2+3^2 = 5^2+1^2+1^2 = 27.
%t A063691 r[n_] := Reduce[ x>0 && y>0 && z>0 && x^2 + y^2 + z^2 == n, {x, y, z}, Integers]; a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === Or, Length[rn], True, 1]; Table[a[n], {n, 0, 89}](* _Jean-François Alcover_, May 10 2012 *)
%t A063691 (EllipticTheta[3, 0, x] - 1)^3/8 + O[x]^100 // CoefficientList[#, x]& (* _Jean-François Alcover_, Jul 30 2017 *)
%Y A063691 Sequence without zeros: A014465.
%Y A063691 Cf. A063725, A063730, A211639 (partial sums).
%Y A063691 Column k=3 of A337165.
%K A063691 easy,nice,nonn
%O A063691 0,7
%A A063691 Andrew A. Doroshev (andy(AT)ip.rsu.ru), Aug 23 2001