A014203 - OEIS

A014203

a(n) = Sum_{i,j,k in Z and i^2 + j^2 + k^2 <= n} i^2 + j^2 + k^2.

0, 6, 30, 54, 78, 198, 342, 342, 438, 708, 948, 1212, 1308, 1620, 2292, 2292, 2388, 3204, 3852, 4308, 4788, 5796, 6324, 6324, 6900, 7650, 9522, 10386, 10386, 12474, 13914, 13914, 14298, 15882, 17514, 19194, 20274, 21162, 23898, 23898, 24858, 28794, 30810, 31842, 32898, 36138, 38346, 38346

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OFFSET

0,2

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

LINKS

Table of n, a(n) for n=0..47.

FORMULA

G.f.: (x/(1-x)) * d/dx(theta_3(x)^3). - Seiichi Manyama, Jul 11 2024

EXAMPLE

Take n=3 say. The sum contains the terms 0 (once) from (0,0,0); 1 (6 times) from (+-1,0,0); 2 (12 times) from (+-1, +-1, 0); and 3 (8 times) from (+-1, +-1, +-1); for a total of 0+6+24+24=54.

PROG

(PARI) my(N=50, x='x+O('x^N)); concat(0, Vec(x/(1-x)*deriv((1+2*sum(k=1, sqrtint(N), x^k^2))^3))) \\ Seiichi Manyama, Jul 11 2024

CROSSREFS

Cf. A000122, A005875.

Sequence in context: A110175 A249676 A259752 * A044083 A377992 A239978

Adjacent sequences: A014200 A014201 A014202 * A014204 A014205 A014206

KEYWORD

nonn

AUTHOR

E.V. Flynn (evflynn(AT)liverpool.ac.uk)

EXTENSIONS

More terms from James A. Sellers, Feb 05 2000

STATUS

approved