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A281352
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Number of ways to write a nonnegative rational integer n as a sum of three squares in the ring of integers of Q(sqrt(3)).
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2
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1, 6, 12, 14, 30, 48, 36, 48, 84, 86, 48, 96, 86, 96, 96, 144, 126, 192, 108, 96, 192, 240, 96, 288, 252, 150, 144, 158, 192, 432, 240, 144, 372, 288, 96, 384, 446, 192, 288, 480, 336, 384, 288, 288, 528, 432, 192, 480, 374, 294, 300, 576, 384, 720, 324, 384
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OFFSET
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0,2
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COMMENTS
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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(3)] of Q(sqrt(3)).
This is the same as solving the system of equations
n = (a^2 + b^2 + c^2) + 3*(d^2 + e^2 + f^2)
ad + be + cf = 0
in rational integers.
According to Cohn (1961), the class number of Q(sqrt(3), sqrt{-n}) always divides a(n).
Let O=Z[sqrt(3)] denote the ring of integers of Q(sqrt(3)). Note that the equation 7=x^2+y^2+z^2 has no solutions in integers, but has 48 solutions in O. For example, 7=1^2+(sqrt(3))^2+(sqrt(3))^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)?
For which n is it true that r_3(n) divides a(n)?
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LINKS
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EXAMPLE
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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)=12, because the only solutions are (x,y,z)=(+-1,+-1,0),(0,+-1,+-1),(+-1,0,+-1);
a(3)=14, because the only solutions are (x,y,z)=(+-1,+-1,+-1),(+-sqrt(3),0,0),(0,+-sqrt(3),0),(0,0,+-sqrt(3));
a(4)=30, etc.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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