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
KEYWORD
nonn,easy
AUTHOR
Anton Mosunov, Jan 22 2017
STATUS
approved