A000605 Number of points of norm <= n in cubic lattice.

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

Author

N. J. A. Sloane

Keywords

• nonn

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).

Crossrefs

Column k=3 of A302997.

Cf. A117609 (number of lattice points inside the ball x^2+y^2+z^2 <= n).

Programs

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 */

Mathematica

Table[Sum[SquaresR[3, k], {k, 0, n^2}], {n, 0, 37}]

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 */

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

Extensions

More terms from David W. Wilson, May 22 2000