A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

1, 7, 19, 27, 33, 57, 81, 81, 93, 123, 147, 171, 179, 203, 251, 251, 257, 305, 341, 365, 389, 437, 461, 461, 485, 515, 587, 619, 619, 691, 739, 739, 751, 799, 847, 895, 925, 949, 1021, 1021, 1045, 1141, 1189, 1213, 1237, 1309, 1357, 1357, 1365, 1419, 1503

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

Formula

a(n) ~ (4/3)*Pi*n^1.5 ~ A210639(n).

a(n) = A122510(3,n). - R. J. Mathar, Apr 21 2010

G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*sum(k>=1, q^(k^2) ). - Joerg Arndt, Apr 08 2013

Example

a(2) = 1 + 6 + 12 = 19, since (0,0,0) and (0, 0, +-1) and cyclic permutations (for a total of 6 points), and +-(0, 1, +-1) and cyclic permutations (for a total 12 points) are inside or on x^2 + y^2 + z^2 = 2.

Mathematica

Table[Sum[SquaresR[3, k], {k, 0, n}], {n, 0, 50}] (* T. D. Noe, Apr 08 2006, revised Sep 27 2011 *)

Prog

(PARI) A117609(n)=sum(x=0, sqrtint(n), (sum(y=1, sqrtint(t=n-x^2), 1+2*sqrtint(t-y^2))*2+sqrtint(t)*2+1)*2^(x>0)) \\ M. F. Hasler, Mar 26 2012
(PARI) q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)) /* Joerg Arndt, Apr 08 2013 */

Crossrefs

Partial sums of A005875.

Cf. A000605 (number of points of norm <= n in cubic lattice).

Cf. A210639, A000092 and references therein.

Sequence in context: A342160 A175376 A175366 * A122072 A261338 A352338

Adjacent sequences: A117606 A117607 A117608 * A117610 A117611 A117612

Keyword

nonn

Author

John L. Drost, Apr 06 2006

Status

approved