A005874 Theta series of hexagonal close-packing with respect to triangle between tetrahedra.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

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

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]

Extensions

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