A002102 Number of nonnegative solutions to x^2 + y^2 + z^2 = n.

1, 3, 3, 1, 3, 6, 3, 0, 3, 6, 6, 3, 1, 6, 6, 0, 3, 9, 6, 3, 6, 6, 3, 0, 3, 9, 12, 4, 0, 12, 6, 0, 3, 6, 9, 6, 6, 6, 9, 0, 6, 15, 6, 3, 3, 12, 6, 0, 1, 9, 15, 6, 6, 12, 12, 0, 6, 6, 6, 9, 0, 12, 12, 0, 3, 18, 12, 3, 9, 12, 6, 0, 6, 9, 18, 7, 3, 12, 6, 0, 6, 15, 9, 9, 6, 12, 15, 0, 3, 21, 18, 6, 0, 6

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 48.
• H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Crossrefs

First differences of A000606.

Programs

Mathematica

a[n_] := Module[{x, y, z, c}, For[x=c=0, x^2<=n, x++, For[y=0, x^2+y^2<=n, y++, If[IntegerQ[Sqrt[n-x^2-y^2]], c++ ]]]; c]
CoefficientList[Series[Sum[q^n^2, {n, 0, 12}], {q, 0, 150}]^3, q]

PARI

Vec(sum(k=0,9,x^(k^2),O(x^100))^3) \\ Charles R Greathouse IV, Jun 13 2012

Formula

Coefficient of q^k in (1/8)*(1 + theta_3(0, q))^3, or coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^3.

Extensions

More terms from Dean Hickerson, Oct 07 2001