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%I #30 Dec 28 2023 01:51:56
%S 1,3,0,6,3,0,0,6,0,6,0,0,6,6,0,0,3,0,0,6,0,12,0,0,0,3,0,6,6,0,0,6,0,0,
%T 0,0,6,6,0,12,0,0,0,6,0,0,0,0,6,9,0,0,6,0,0,0,0,12,0,0,0,6,0,12,3,0,0,
%U 6,0,0,0,0,0,6,0,6,6,0,0,6,0,6,0,0,12,0,0,0,0,0,0,12,0,12,0,0,0,6,0,0
%N Expansion of theta series of hexagonal net with respect to a node.
%C The hexagonal net is the familiar 2-dimensional honeycomb (not a lattice) in which each node has 3 neighbors.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A. F. Wells, Structural Inorganic Chemistry, Oxford, 5th ed., 1984; see Fig. 3.9(a.1).
%H T. D. Noe, <a href="/A113062/b113062.txt">Table of n, a(n) for n = 0..1000</a>
%H George L. Hall, <a href="http://dx.doi.org/10.1063/1.527833">Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]</a>. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
%H N. J. A. Sloane, <a href="http://dx.doi.org/10.1063/1.527472">Theta series and magic numbers for diamond and certain ionic crystal structures</a>, J. Math. Phys. 28 (1987), 1653-1657.
%F Moebius transform is period 9 sequence [ 3, -3, 3, 3, -3, -3, 3, -3, 0, ...].
%F Expansion of a(q^3) + c(q^3) in powers of q where a(), c() are cubic AGM theta functions. - _Michael Somos_, Aug 15 2006
%F For n>0, a(n) = 3*b(n) where b(n)=A113063(n) is multiplicative and b(p^e) = 2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
%F a(3*n + 2) = 0. a(3*n + 1) = A005882(n) = A033685(3*n + 1) = -A005928(3*n + 1). a(3*n) = A004016(n) = A005928(3*n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - _Amiram Eldar_, Dec 28 2023
%e G.f. = 1 + 3*q + 6*q^3 + 3*q^4 + 6*q^7 + 6*q^9 + 6*q^12 + 6*q^13 + 3*q^16 + ...
%t a[0] = 1; a[n_] := 3*DivisorSum[n, {0, 1, -1, 1, 1, -1, -1, 1, -1}[[Mod[#, 9]+1]]&]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 04 2015, after 1st PARI script *)
%o (PARI) {a(n) = if( n<1, n==0, 3 * sumdiv(n, d, [ 0, 1, -1, 1, 1, -1, -1, 1, -1][d%9+1]))};
%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 3 * prod(k=1, matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==3, 2, if(p%6==1, e+1, !(e%2))))))};
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); abs( polcoeff( eta(x + A)^3 / eta(x^3 + A), n)))};
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 6 * x * eta(x^9 + A)^3) / eta(x^3 + A), n))}; /* _Michael Somos_, Aug 15 2006 */
%Y Cf. A004016, A005882, A005928, A033685, A113063, A275486.
%Y Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Oct 13 2005
%E Definition corrected _Michael Somos_, Oct 17 2005
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