A281484 Number of ways to write a nonnegative rational integer n as a sum of three squares in the ring of integers of Q(sqrt 7).

1, 6, 12, 8, 6, 24, 24, 6, 36, 54, 24, 48, 56, 24, 60, 48, 54, 144, 60, 24, 120, 56, 48, 144, 72, 102, 120, 128, 6, 144, 96, 48, 180, 192, 48, 168, 198, 48, 168, 144, 72, 384, 72, 48, 288, 216, 96, 336, 152, 150, 204, 96, 120, 288, 192, 96, 372, 240, 96, 360

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 7] of Q(sqrt 7).

This is the same as solving the system of equations

n = (a^2 + b^2 + c^2) + 7*(d^2 + e^2 + f^2)

ad + be + cf = 0

in rational integers.

Links

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

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(7)=6, because the only solutions are (x,y,z)=(+-sqrt 7,0,0),(0,+-sqrt 7,0),(0,0,+- sqrt 7)
Let O denote the ring of integers Z[sqrt 7] of Q(sqrt 7). 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=(sqrt 7)^2.
What is the relationship between the class number of Q(sqrt 7, sqrt(-n)) and a(n)?
Is there a generating function for a(n)?

Crossrefs

Cf. A280802, A281352.

Sequence in context: A236933 A236927 A236932 * A236931 A028659 A236930

Adjacent sequences: A281481 A281482 A281483 * A281485 A281486 A281487

Keyword

nonn,easy

Author

Anton Mosunov, Jan 22 2017

Status

approved