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The number of solutions to x^2 + y^2 + 2*z^2 = n in positive integers x,y,z.
3

%I #16 Feb 16 2024 06:32:45

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

%F a(n) = ( A014455(n) - 2*A033715(n) - A004018(n) + A000122(n/2) + 2*A000122(n) - A000007(n) )/8.

%F G.f.: T(x)^2 * T(x^2) where T(x) = sum(k>=1, x^(k^2)). [_Joerg Arndt_, Oct 01 2012]

%o (PARI)

%o N=166; x='x+O('x^N);

%o T(x)=sum(k=1, 1+sqrtint(N), x^(k*k) );

%o gf=T(x)^2 * T(x^2);

%o v=Vec('a0 + gf ); v[1]=0; v

%o /* _Joerg Arndt_, Oct 01 2012 */

%Y Cf. A156384

%K nonn

%O 0,8

%A _Max Alekseyev_, Sep 29 2012