A005872 - OEIS

A005872

Theta series of hexagonal close-packing with respect to octahedral hole.
(Formerly M4040)

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

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OFFSET

0,4

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

FORMULA

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

EXAMPLE

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

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 *)

PROG

(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 */

CROSSREFS

Cf. A298931.

Sequence in context: A200214 A294887 A331979 * A035322 A270031 A284105

Adjacent sequences: A005869 A005870 A005871 * A005873 A005874 A005875

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved