A280802 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).

1, 6, 18, 32, 42, 48, 80, 96, 90, 54, 144, 96, 176, 144, 192, 192, 186, 192, 162, 288, 336, 192, 240, 288, 368, 150, 432, 320, 384, 144, 384, 576, 378, 384, 576, 384, 378, 240, 720, 384, 720, 384, 576, 480, 528, 432, 576, 960, 752, 486, 450, 384, 1008, 432

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 2] of Q(sqrt 2).

This is the same as solving the system of equations

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

ad + be + cf = 0

in rational integers.

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

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.

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)?

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?

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

Links

Table of n, a(n) for n=0..53.

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.

D. Ye, Representations of integers by certain 2k-ary binary forms, arXiv:1607.00088 [math.NT], 2016.

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)=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)
a(3)=32, etc.

Crossrefs

Cf. A005875.

Sequence in context: A096286 A256256 A344596 * A124353 A232336 A153126

Adjacent sequences: A280799 A280800 A280801 * A280803 A280804 A280805

Keyword

nonn,easy

Author

Anton Mosunov, Jan 08 2017

Status

approved