1,3

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Also the number of triangles with vertices on points of the hexagonal lattice that have area equal to n/2. - Amihay Hanany, Oct 15 2009 [Here the area is measured in the units of the lattice unit cell area; since the number of the triangles of different shapes with the same half-integral area is infinite, the triangles are probably counted up to the equivalence relation defined in the Davey, Hanany and Rak-Kyeong Seong paper. Also, this comment probably belongs to A003051, not here. - Andrey Zabolotskiy, Mar 10 2018 and Jul 04 2019]

Also number of 2n-vertex connected cubic vertex-transitive graphs which are Cayley graphs for a dihedral group [Potočnik et al.]. - N. J. A. Sloane, Apr 19 2014

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

T. D. Noe, Table of n, a(n) for n = 1..1000

A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.

A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]

M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).

J. Davey, A. Hanany and Rak-Kyeong Seong, Counting Orbifolds, arXiv:1002.3609 [hep-th], 2010.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.

Primož Potočnik, Pablo Spiga and Gabriel Verret, A census of small connected cubic vertex-transitive graphs (See the sub-page Table.html, column headed "Dihedrants"). - N. J. A. Sloane, Apr 19 2014

Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6)*n*Product_{i=1..w} (1 + 1/p_i) + (C_1)/3 + 2^(w-2+C_2),

where C_1 = 0 if 2|n or 9|n, = Product_{i=1..w, p_i > 3} (1 + Legendre(p_i, 3)) otherwise,

and C_2 = 2 if n == 0 (mod 8), 1 if n == 1, 3, 4, 5, 7 (mod 8), 0 if n == 2, 6 (mod 8).

For n = 6 the 3 primitive triples are 510, 411, 321.

Join[{1}, Table[p=Transpose[FactorInteger[n]][[1]]; If[Mod[n, 2]==0 || Mod[n, 9]==0, c1=0, c1=Product[(1+JacobiSymbol[p[[i]], 3]), {i, Length[p]}]]; c2={2, 1, 0, 1, 1, 1, 0, 1}[[1+Mod[n, 8]]]; n*Product[(1+1/p[[i]]), {i, Length[p]}]/6+c1/3+2^(Length[p]-2+c2), {n, 2, 100}]] (* T. D. Noe, Oct 18 2009 *)

Cf. A003051 (not only primitive sublattices), A001615, A006984, A007997, A048259, A054345, A154272, A157235.

Sequence in context: A078342 A177903 A107325 * A070868 A380335 A342219

Adjacent sequences: A003047 A003048 A003049 * A003051 A003052 A003053

nonn,nice

approved