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

0,2

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.

Links

Table of n, a(n) for n=0..99.

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

Example

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

Sequence in context: A349956 A104564 A160764 * A306249 A064656 A270776

Adjacent sequences: A156381 A156382 A156383 * A156385 A156386 A156387

Keyword

nonn

Author

R. H. Hardin Feb 09 2009

Extensions

More general definition from Max Alekseyev, Sep 29 2012

Status

approved