A004012 Theta series of hexagonal close-packing. (Formerly M4817)

1, 0, 0, 12, 0, 0, 6, 0, 2, 18, 0, 12, 6, 0, 0, 12, 0, 12, 6, 6, 12, 24, 6, 0, 0, 12, 0, 12, 0, 24, 12, 12, 2, 12, 6, 24, 6, 12, 0, 24, 0, 12, 0, 6, 24, 12, 12, 24, 6, 12, 0, 24, 0, 24, 18, 12, 12, 24, 0, 12, 0, 12, 0, 36, 0, 24, 12, 18, 12, 24, 12, 48, 2, 0, 0, 36, 0, 0, 24, 12, 12

Offset

0,4

Comments

The theta series of even layers is a(q^3) * theta_3(q^8) and of odd layers is c(q^3) * theta_2(q^8). - Michael Somos, Aug 15 2006

The Cartesian coordinates of the points in the packing are given by HCP(i, j, k) =

(i + (j + m)/2, (3*j + m)/sqrt(12), sqrt(2/3)*k) where, m=mod(k, 2) and i, j, k are integers. - Michael Somos, Feb 04 2019

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

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

Links

T. D. Noe, Table of n, a(n) for n = 0..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

L. V. Woodcock, Entropy difference between the face-centred cubic and hexagonal close-packed crystal structures, Nature, Jan 09 1997, pp. 141-143, esp. p. 143.

Formula

{t3(8z/3) - t2(8z/3)/2} * {t3(z)t3(3z) + t2(z)t2(3z)} + (1/2)*t2(8z/3) * {t3(z/3)t3(z) + t2(z/3)t2(z)}, where t3=theta_3, t2=theta_2.

Expansion of a(x^3) * phi(x^8) + 2*x^2 * c(x^3) * psi(x^16) in powers of x where a(), c() are cubic AGM theta functions and phi(), psi() are Ramanujan theta functions.

a(n) is the number of integer solutions [i, j, k] to n = 2*i^2 + (j^2 + j*k + k^2) / 3 where j, k == mod(i, 2) (mod 3). - Michael Somos, Jul 19 2014

G.f.: Sum_{i, j, k in Z} x^(8*i^2 + 3*(j^2 + j*k + k^2)) * (1 + x^(3 + 8*i + 3*j + 3*k)). - Michael Somos, Jul 19 2014

Example

G.f. = 1 + 12*x^3 + 6*x^6 + 2*x^8 + 18*x^9 + 12*x^11 + 6*x^12 + 12*x^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 + 9 x^3 QPochhammer[ x^27]^3) / QPochhammer[ x^9] EllipticTheta[ 3, 0, x^8] + 3 x QPochhammer[ x^9]^3 / QPochhammer[ x^3]  EllipticTheta[ 2, 0, x^8], {x, 0, n}]; (* Michael Somos, Jul 19 2014 *)
a[ n_] := SeriesCoefficient[ 6 x^3 QPochhammer[ x^32]^2 / ( QPochhammer[ x^3] QPochhammer[ x^16]) + 2 EllipticTheta[ 3, 0, x^3]  EllipticTheta[ 3, 0, x^8] EllipticTheta[ 3, 0, x^9] - EllipticTheta[ 4, 0, x^3]   EllipticTheta[ 4, 0, x^8] EllipticTheta[ 4, 0, x^9], {x, 0, n}]; (* Michael Somos, Jul 19 2014 *)

Prog

(PARI) {a(n) = my(A, A0, A1); if( n<0, 0, A = x * O(x^n); A1 = x^3 * eta(x^9 + A)^3 * eta(x^32 + A)^2 / (eta(x^3 + A) * eta(x^16 + A)); A0 = sum(k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A) * sum(k=1, sqrtint(n\8), 2 * x^(8*k^2), 1 + A) * sum(k=1, sqrtint(n\9), 2 * x^(9*k^2), 1 + A); polcoeff( 2*A0 + 6*A1 - subst(A0, x, -x), n))}; /* Michael Somos, Aug 03 2006 */

Crossrefs

Sequence in context: A225341 A369349 A368816 * A360223 A270256 A072837

Adjacent sequences: A004009 A004010 A004011 * A004013 A004014 A004015

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane

Status

approved