A281352 - OEIS

A281352

Number of ways to write a nonnegative rational integer n as a sum of three squares in the ring of integers of Q(sqrt(3)).

1, 6, 12, 14, 30, 48, 36, 48, 84, 86, 48, 96, 86, 96, 96, 144, 126, 192, 108, 96, 192, 240, 96, 288, 252, 150, 144, 158, 192, 432, 240, 144, 372, 288, 96, 384, 446, 192, 288, 480, 336, 384, 288, 288, 528, 432, 192, 480, 374, 294, 300, 576, 384, 720, 324, 384

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OFFSET

0,2

COMMENTS

a(n) is the number of solutions to the equation n = x^2 + y^2 + z^2 with x, y, z in the ring of integers Z[sqrt(3)] of Q(sqrt(3)).

This is the same as solving the system of equations

n = (a^2 + b^2 + c^2) + 3*(d^2 + e^2 + f^2)

ad + be + cf = 0

in rational integers.

According to Cohn (1961), the class number of Q(sqrt(3), sqrt{-n}) always divides a(n).

Let O=Z[sqrt(3)] denote the ring of integers of Q(sqrt(3)). Note that the equation 7=x^2+y^2+z^2 has no solutions in integers, but has 48 solutions in O. For example, 7=1^2+(sqrt(3))^2+(sqrt(3))^2.

Let theta_3(q)=1+2q+2q^4+... be the 3rd Jacobi theta function. It is widely known that theta_3(q)^3 is the generating function for the number of rational integer solutions r_3(n) to n=x^2+y^2+z^2.

Is there a generating function for a(n)?

For which n is it true that r_3(n) divides a(n)?

LINKS

Anton Mosunov, Table of n, a(n) for n = 0..1000

H. Cohn, Calculation of class numbers by decomposition into 3 integral squares in the fields of 2^{1/2} and 3^{1/2}, American Journal of Mathematics 83 (1), pp. 33-56, 1961.

Wikipedia, Ring of integers.

EXAMPLE

a(0)=1, because the equation 0 = x^2 + y^2 + z^2 has a single solution (x,y,z)=(0,0,0);

a(1)=6, because the only solutions are (x,y,z)=(+-1,0,0),(0,+-1,0),(0,0,+-1);

a(2)=12, because the only solutions are (x,y,z)=(+-1,+-1,0),(0,+-1,+-1),(+-1,0,+-1);

a(3)=14, because the only solutions are (x,y,z)=(+-1,+-1,+-1),(+-sqrt(3),0,0),(0,+-sqrt(3),0),(0,0,+-sqrt(3));

a(4)=30, etc.