A156384 The number of solutions to x^2 + y^2 + 2*z^2 = n in nonnegative integers x,y,z.

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, 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, 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

Offset: 0

Author

R. H. Hardin Feb 09 2009

Keywords

• nonn

Comments

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:

z^2 x^2 y^2 z^2

y^2 z^2 z^2 x^2

x^2 z^2 z^2 y^2

z^2 y^2 x^2 z^2

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

Examples

			All matrices for n=9:
...0.0.9.0...0.9.0.0...4.0.1.4...4.1.0.4
...9.0.0.0...0.0.0.9...1.4.4.0...0.4.4.1
...0.0.0.9...9.0.0.0...0.4.4.1...1.4.4.0
...0.9.0.0...0.0.9.0...4.1.0.4...4.0.1.4
		

Crossrefs

Cf. A213024

Formula

a(n) = ( A014455(n) + 2*A033715(n) + A004018(n) + A000122(n/2) + 2*A000122(n) + A000007(n) )/8. - Max Alekseyev, Sep 29 2012

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

Extensions

More general definition from Max Alekseyev, Sep 29 2012