A002102 - OEIS

A002102

Number of nonnegative solutions to x^2 + y^2 + z^2 = n.
(Formerly M2265 N0895)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

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.

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]

PROG

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

CROSSREFS

First differences of A000606.

Sequence in context: A332547 A302694 A245668 * A332552 A047655 A078685

Adjacent sequences: A002099 A002100 A002101 * A002103 A002104 A002105

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Dean Hickerson, Oct 07 2001

STATUS

approved