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A000605
Number of points of norm <= n in cubic lattice.
(Formerly M4406 N1860)
12
1, 7, 33, 123, 257, 515, 925, 1419, 2109, 3071, 4169, 5575, 7153, 9171, 11513, 14147, 17077, 20479, 24405, 28671, 33401, 38911, 44473, 50883, 57777, 65267, 73525, 82519, 91965, 101943, 113081, 124487, 137065, 150555, 164517, 179579, 195269, 212095
OFFSET
0,2
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
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
W. Fraser and C. C. Gotlieb, A calculation of the number of lattice points in the circle and sphere, Math. Comp., 16 (1962), 282-290.
Z. C. Holden, R. M. Richard, J. M. Herbert, Periodic boundary conditions for QM/MM calculations: Ewald summation for extended Gaussian basis sets, The Journal of Chemical Physics, J. Chem. Phys. 139, 244108 (2013).
FORMULA
a(n) = A117609(n^2). - R. J. Mathar, Apr 21 2010
a(n) = [x^(n^2)] theta_3(x)^3/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018
MATHEMATICA
Table[Sum[SquaresR[3, k], {k, 0, n^2}], {n, 0, 37}]
PROG
(C)
int A000605(int i)
{
const int ring = i*i;
int result = 0;
for (int a = -i; a <= i; a++)
for (int b = -i; b <= i; b++)
for (int c = -i; c <= i; c++)
if ( ring >= a*a+b*b+c*c ) result++;
return result;
} /* Oskar Wieland, Apr 08 2013 */
(PARI)
N=66; q='q+O('q^(N^2));
t=Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)); /* A117609 */
vector(sqrtint(#t), n, t[(n-1)^2+1])
/* Joerg Arndt, Apr 08 2013 */
CROSSREFS
Column k=3 of A302997.
Cf. A117609 (number of lattice points inside the ball x^2+y^2+z^2 <= n).
Sequence in context: A256860 A221036 A338232 * A215054 A350643 A114014
KEYWORD
nonn
EXTENSIONS
More terms from David W. Wilson, May 22 2000
STATUS
approved