%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