

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).


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
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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 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).


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. 3356, 1961.
D. Ye, Representations of integers by certain 2kary 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.
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+(1sqrt 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)?


CROSSREFS

Cf. A005875.
Sequence in context: A017593 A096286 A256256 * A124353 A232336 A153126
Adjacent sequences: A280799 A280800 A280801 * A280803 A280804 A280805


KEYWORD

nonn,easy


AUTHOR

Anton Mosunov, Jan 08 2017


STATUS

approved



