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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 (list; graph; refs; listen; history; text; internal format)
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. 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.

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)?

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

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)