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

0,8

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of squares

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

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 *)

Prog

(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

Crossrefs

Cf. A063691, A063725.

Column k=4 of A337165.

Sequence in context: A212044 A290335 A341795 * A355945 A131431 A240664

Adjacent sequences: A063727 A063728 A063729 * A063731 A063732 A063733

Keyword

nonn

Author

N. J. A. Sloane, Aug 23 2001

Status

approved