A063730 Number of solutions to w^2 + x^2 + y^2 + z^2 = n in positive integers.

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

Offset: 0

Author

N. J. A. Sloane, Aug 23 2001

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A063691, A063725.

Column k=4 of A337165.

Programs

Mathematica

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 *)

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

Formula

G.f.: (Sum_{m>=1} x^(m^2))^4.

a(n) = ( A000118(n) - 4*A005875(n) + 6*A004018(n) - 4*A000122(n) + A000007(n) )/16. - Max Alekseyev, Sep 29 2012

G.f.: ((theta_3(q) - 1)/2)^4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018