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

Author

John L. Drost, Apr 06 2006

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

Partial sums of A005875.

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

Cf. A210639, A000092 and references therein.

Cf. A057655.

Programs

Mathematica

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

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 */

Python

# uses Python code for A057655
from math import isqrt
def A117609(n): return A057655(n)+(sum(A057655(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

Formula

a(n) ~ (4/3)*Pi*n^(3/2) ~ 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

a(n^2) = A000605(n). - R. J. Mathar, Aug 03 2025