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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
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
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
Sequence in context: A236933 A236927 A236932 * A236931 A028659 A236930
KEYWORD
nonn,easy
AUTHOR
Anton Mosunov, Jan 22 2017
STATUS
approved