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A213024 The number of solutions to x^2 + y^2 + 2*z^2 = n in positive integers x,y,z.

%I A213024 #16 Feb 16 2024 06:32:45
%S A213024 0,0,0,0,1,0,0,2,0,0,2,0,2,2,0,2,1,0,2,2,2,2,2,2,0,2,2,2,6,0,0,4,0,2,
%T A213024 4,2,3,4,2,2,2,0,6,4,2,4,0,4,2,4,2,0,8,2,2,6,0,2,8,2,6,4,0,6,1,0,4,6,
%U A213024 4,4,6,2,2,6,2,4,8,4,0,4,2,2,10,4,6,4,2,6,2,2,8,6,6,6,0,2,0,8,6,2,9
%N A213024 The number of solutions to x^2 + y^2 + 2*z^2 = n in positive integers x,y,z.
%F A213024 a(n) = ( A014455(n) - 2*A033715(n) - A004018(n) + A000122(n/2) + 2*A000122(n) - A000007(n) )/8.
%F A213024 G.f.: T(x)^2 * T(x^2) where T(x) = sum(k>=1, x^(k^2)). [_Joerg Arndt_, Oct 01 2012]
%o A213024 (PARI)
%o A213024 N=166; x='x+O('x^N);
%o A213024 T(x)=sum(k=1, 1+sqrtint(N), x^(k*k) );
%o A213024 gf=T(x)^2 * T(x^2);
%o A213024 v=Vec('a0 + gf );  v[1]=0;  v
%o A213024 /* _Joerg Arndt_, Oct 01 2012 */
%Y A213024 Cf. A156384
%K A213024 nonn
%O A213024 0,8
%A A213024 _Max Alekseyev_, Sep 29 2012