A004012 Theta series of hexagonal close-packing.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Examples

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

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.

Programs

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 *)

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 */

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