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%I M4817 #26 Mar 12 2021 22:24:41
%S 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,
%T 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,
%U 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
%N Theta series of hexagonal close-packing.
%C 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
%C The Cartesian coordinates of the points in the packing are given by HCP(i, j, k) =
%C (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
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A004012/b004012.txt">Table of n, a(n) for n = 0..5000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H L. V. Woodcock, <a href="http://dx.doi.org/10.1038/385141a0">Entropy difference between the face-centred cubic and hexagonal close-packed crystal structures</a>, Nature, Jan 09 1997, pp. 141-143, esp. p. 143.
%F {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.
%F 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.
%F 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
%F 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
%e G.f. = 1 + 12*x^3 + 6*x^6 + 2*x^8 + 18*x^9 + 12*x^11 + 6*x^12 + 12*x^15 + ...
%t 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 *)
%t 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 *)
%o (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 */
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
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