A005872 Theta series of hexagonal close-packing with respect to octahedral hole.

0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 18, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 24, 0, 6, 0, 0, 0, 36, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 0, 0, 18

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			G.f. = 6*q^3 + 6*q^9 + 6*q^11 + 12*q^15 + 6*q^17 + 12*q^23 + 18*q^27 + ... - _Michael Somos_, Jul 06 2018
		

References

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

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..999

Crossrefs

Cf. A298931.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 6 x^3 QPochhammer[ x^16]^2 QPochhammer[ x^18]^3 / (QPochhammer[ x^6] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 06 2018 *)

PARI

{a(n) = my(A, m); if( n<3 || n%2==0, 0, m = n\2 - 1; A = x * O(x^m); 6 * polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), m))}; /* Michael Somos, Jul 06 2018 */

Formula

a(2*n) = 0. a(2*n + 3) = 6*A298931(n). - Michael Somos, Jul 06 2018