A000604 Number of nonnegative solutions to x^2 + y^2 + z^2 <= n^2. (Formerly M3419 N1383)

1, 4, 11, 29, 54, 99, 163, 239, 344, 486, 648, 847, 1069, 1355, 1680, 2046, 2446, 2911, 3443, 4022, 4662, 5395, 6145, 6998, 7913, 8913, 10006, 11194, 12437, 13751, 15216, 16710, 18361, 20123, 21950, 23919, 25956, 28150, 30415, 32876, 35385, 38049, 40876

Offset

0,2

References

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..500

Formula

a(n) = [x^(n^2)] (1 + theta_3(x))^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

Mathematica

a[n_] := Sum[Boole[x^2 + y^2 + z^2 <= n^2], {x, 0, n}, {y, 0, Sqrt[n^2 - x^2]}, {z, 0, Sqrt[n^2 - x^2 - y^2]}]; A000604 = Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* Jean-François Alcover, Feb 10 2016 *)

Crossrefs

Column k=3 of A302998.

Cf. A000606.

Sequence in context: A152689 A217918 A360447 * A153876 A036881 A275012

Adjacent sequences: A000601 A000602 A000603 * A000605 A000606 A000607

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, May 22 2000

Status

approved