A217219 Theta series of planar hexagonal net (honeycomb) with respect to deep hole.

0, 6, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0

Offset: 0

Author

N. J. A. Sloane, Oct 05 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Crossrefs

Cf. A033685, A045833, A275486.

Programs

Mathematica

terms = 105; s = 6 q QPochhammer[q^9]^3/QPochhammer[q^3] + O[q]^(terms+5); CoefficientList[s, q][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017, after Michael Somos *)
CoefficientList[Series[6 q QPochhammer[q^9]^3/QPochhammer[q^3], {q, 0, 100}], q] (* G. C. Greubel, Aug 10 2018 *)

PARI

my(q='q+O('q^100)); concat([0], Vec(6*q*eta(q^9)^3/eta(q^3))) \\ G. C. Greubel, Aug 10 2018

Formula

See A033685, A045833.

a(n) = 2*A033685(n) = 6*A045833(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Oct 13 2022

Extensions

Name edited by Andrey Zabolotskiy, Jun 21 2022