A113062 Expansion of theta series of hexagonal net with respect to a node.

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, 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, 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

Offset

0,2

Comments

The hexagonal net is the familiar 2-dimensional honeycomb (not a lattice) in which each node has 3 neighbors.

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

References

A. F. Wells, Structural Inorganic Chemistry, Oxford, 5th ed., 1984; see Fig. 3.9(a.1).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

George L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From N. J. A. Sloane, Dec 18 2012

N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Formula

Moebius transform is period 9 sequence [ 3, -3, 3, 3, -3, -3, 3, -3, 0, ...].

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

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).

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).

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

Example

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 + ...

Mathematica

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 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 3 * sumdiv(n, d, [ 0, 1, -1, 1, 1, -1, -1, 1, -1][d%9+1]))};
(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))))))};
(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)))};
(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 */

Crossrefs

Cf. A004016, A005882, A005928, A033685, A113063, A275486.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A339416 A226535 A005928 * A259659 A005889 A095712

Adjacent sequences: A113059 A113060 A113061 * A113063 A113064 A113065

Keyword

nonn,easy

Author

Michael Somos, Oct 13 2005

Extensions

Definition corrected Michael Somos, Oct 17 2005

Status

approved