A063691 Number of solutions to x^2 + y^2 + z^2 = n in positive integers.

0, 0, 0, 1, 0, 0, 3, 0, 0, 3, 0, 3, 1, 0, 6, 0, 0, 3, 3, 3, 0, 6, 3, 0, 3, 0, 6, 4, 0, 6, 6, 0, 0, 6, 3, 6, 3, 0, 9, 0, 0, 9, 6, 3, 3, 6, 6, 0, 1, 6, 6, 6, 0, 6, 12, 0, 6, 6, 0, 9, 0, 6, 12, 0, 0, 6, 12, 3, 3, 12, 6, 0, 3, 3, 12, 7, 3, 12, 6, 0, 0, 12, 3, 9, 6, 0, 15, 0, 3, 15

Offset

0,7

Links

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

Formula

G.f.: (Sum_{m>=1} x^(m^2))^3.

Example

a(5)=0;
a(6)=3 because 1^2+1^2+2^2 = 1^2+2^2+1^2 = 2^2+1^2+1^2 = 6;
a(27)=4 because 1^2+1^2+5^2 = 1^2+5^2+1^2 = 3^2+3^2+3^2 = 5^2+1^2+1^2 = 27.

Mathematica

r[n_] := Reduce[ x>0 && y>0 && z>0 && x^2 + y^2 + z^2 == n, {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, 89}](* Jean-François Alcover, May 10 2012 *)
(EllipticTheta[3, 0, x] - 1)^3/8 + O[x]^100 // CoefficientList[#, x]& (* Jean-François Alcover, Jul 30 2017 *)

Crossrefs

Sequence without zeros: A014465.

Cf. A063725, A063730, A211639 (partial sums).

Column k=3 of A337165.

Sequence in context: A341794 A033685 A272974 * A359967 A284281 A075874

Adjacent sequences: A063688 A063689 A063690 * A063692 A063693 A063694

Keyword

easy,nice,nonn

Author

Andrew A. Doroshev (andy(AT)ip.rsu.ru), Aug 23 2001

Status

approved