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A063730
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Number of solutions to w^2 + x^2 + y^2 + z^2 = n in positive integers.
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14
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0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 6, 0, 4, 4, 0, 12, 1, 0, 12, 4, 6, 4, 12, 12, 0, 12, 6, 12, 12, 0, 24, 16, 0, 12, 18, 12, 13, 16, 12, 28, 6, 0, 36, 16, 12, 24, 24, 24, 4, 16, 30, 24, 18, 12, 36, 36, 0, 28, 42, 12, 36, 16, 24, 52, 1, 24, 48, 28, 18, 24, 60, 36, 12
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OFFSET
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0,8
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LINKS
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FORMULA
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G.f.: (Sum_{m>=1} x^(m^2))^4.
G.f.: ((theta_3(q) - 1)/2)^4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018
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MATHEMATICA
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r[n_] := Reduce[ w > 0 && x > 0 && y > 0 && z > 0 && w^2 + x^2 + y^2 + z^2 == n, {w, x, y, z}, Integers]; a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === Or, Length[rn], True, 1]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jul 22 2013 *)
a[n_ ] := Length[FindInstance[{n == w^2 + x^2 + y^2 + z^2, w > 0, x > 0, y > 0, z > 0}, {w, x, y, z}, Integers, 10^18]]; (* Michael Somos, Jun 23 2023 *)
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PROG
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(PARI) seq(n)=Vec((sum(k=1, sqrtint(n), x^(k^2)) + O(x*x^n))^4 + O(x*x^n), -(n+1)) \\ Andrew Howroyd, Aug 08 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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