A005874 Theta series of hexagonal close-packing with respect to triangle between tetrahedra. (Formerly M2236)

0, 3, 2, 0, 3, 12, 0, 6, 0, 6, 0, 12, 6, 6, 12, 12, 3, 0, 2, 6, 0, 24, 0, 24, 6, 3, 0, 24, 6, 12, 12, 6, 0, 12, 0, 0, 18, 6, 12, 48, 0, 24, 0, 6, 0, 36, 0, 0, 6, 9, 14, 24, 6, 12, 12, 0, 0, 48, 0, 36, 24, 6, 12, 12, 3, 24, 12, 6, 0, 24, 0, 24, 6, 12, 0, 48, 12

Offset

0,2

Comments

Just take the theta series for the h.c.p. and subtract the coordinates of the center of the triangle from each point. - N. J. A. Sloane, May 18 2021

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000

S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Formula

Sum_{n<=x} a(n)^2 ~ (8*Pi^4/(21*zeta(3))) * x^2. (Choi/Kumchev/Osburn) [Corrected by Vaclav Kotesovec, Oct 25 2015]

Crossrefs

Cf. A004012, A005872, A005873, A005889, A005890.

Sequence in context: A331922 A198826 A088161 * A275622 A129239 A224825

Adjacent sequences: A005871 A005872 A005873 * A005875 A005876 A005877

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Terms a(63) and beyond from Andrey Zabolotskiy, Jun 20 2022

Status

approved