A000605 Number of points of norm <= n in cubic lattice. (Formerly M4406 N1860)

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

T. D. Noe, Table of n, a(n) for n=0..500

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

Adjacent sequences: A000602 A000603 A000604 * A000606 A000607 A000608

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, May 22 2000

Status

approved