A280802 - OEIS

%I #22 Apr 02 2020 14:07:08

%S 1,6,18,32,42,48,80,96,90,54,144,96,176,144,192,192,186,192,162,288,

%T 336,192,240,288,368,150,432,320,384,144,384,576,378,384,576,384,378,

%U 240,720,384,720,384,576,480,528,432,576,960,752,486,450,384,1008,432

%N Number of ways to write a nonnegative rational integer n as a sum of three squares in the ring of integers of Q(sqrt 2).

%C 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 2] of Q(sqrt 2).

%C This is the same as solving the system of equations

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

%C   ad + be + cf = 0

%C in rational integers.

%C According to Cohn (1961), the class number of Q(sqrt 2, sqrt{-n}) always divides a(n).

%C Let O=Z[sqrt 2] denote the ring of integers of Q(sqrt 2). Note that the equation 7=x^2+y^2+z^2 has no solutions in integers, but has 96 solutions in O. For example, 7=1^2+(1+sqrt 2)^2+(1-sqrt 2)^2.

%C 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.

%C Is there a generating function for a(n)?

%C According to Ye (2016), there is a generating function for the number of rational integer solutions of n=(a^2+b^2+c^2)+2*(d^2+e^2+f^2). Is it possible to incorporate the condition ad+be+cf=0?

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

%H H. Cohn, <a href="http://www.jstor.org/stable/2372719">Calculation of class numbers by decomposition into 3 integral squares in the fields of 2^{1/2} and 3^{1/2}</a>, American Journal of Mathematics 83 (1), pp. 33-56, 1961.

%H D. Ye, <a href="https://arxiv.org/abs/1607.00088">Representations of integers by certain 2k-ary binary forms</a>, arXiv:1607.00088 [math.NT], 2016.

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

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

%e a(2)=18, because the only solutions are (x,y,z)=(+-1,+-1,0),(0,+-1,+-1),(+-1,0,+-1),(+-sqrt 2,0,0),(0,+-sqrt 2,0),(0,0,+-sqrt 2)

%e a(3)=32, etc.

%Y Cf. A005875.

%K nonn,easy

%O 0,2

%A _Anton Mosunov_, Jan 08 2017