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A000605
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Number of points of norm <= n in cubic lattice.
(Formerly M4406 N1860)
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11
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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
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OFFSET
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0,2
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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Table[Sum[SquaresR[3, k], {k, 0, n^2}], {n, 0, 37}]
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PROG
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(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 */
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from David W. Wilson, May 22 2000
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STATUS
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approved
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