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The number of solutions to x^2 + y^2 + 2*z^2 = n in nonnegative integers x,y,z.
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%I #17 Aug 01 2018 09:44:30

%S 1,2,2,2,3,2,2,2,2,4,4,2,4,4,0,2,3,4,6,4,4,2,4,2,2,6,4,6,6,2,0,4,2,6,

%T 8,2,7,6,4,2,4,4,6,6,4,6,0,4,4,6,6,4,10,4,6,6,0,6,10,4,6,6,0,6,3,4,8,

%U 8,8,4,6,2,6,10,4,6,10,4,0,4,4,8,14,6,6,8,4,6,4,6,10,6,6,6,0,2,2,12,8,8

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

%C Also, the number of 4X4 matrices composed of squares of integers, symmetric under 90 degree rotation, with all rows summing to n. Such matrices have the form:

%C z^2 x^2 y^2 z^2

%C y^2 z^2 z^2 x^2

%C x^2 z^2 z^2 y^2

%C z^2 y^2 x^2 z^2

%C with x^2 + y^2 + 2*z^2 = n.

%F a(n) = ( A014455(n) + 2*A033715(n) + A004018(n) + A000122(n/2) + 2*A000122(n) + A000007(n) )/8. - _Max Alekseyev_, Sep 29 2012

%F G.f.: (1 + theta_3(q))^2*(1 + theta_3(q^2))/8, where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 01 2018

%e All matrices for n=9:

%e ...0.0.9.0...0.9.0.0...4.0.1.4...4.1.0.4

%e ...9.0.0.0...0.0.0.9...1.4.4.0...0.4.4.1

%e ...0.0.0.9...9.0.0.0...0.4.4.1...1.4.4.0

%e ...0.9.0.0...0.0.9.0...4.1.0.4...4.0.1.4

%Y Cf. A213024

%K nonn

%O 0,2

%A _R. H. Hardin_ Feb 09 2009

%E More general definition from _Max Alekseyev_, Sep 29 2012