A003050 Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.

1, 1, 2, 2, 2, 3, 3, 4, 3, 4, 3, 6, 4, 5, 6, 6, 4, 7, 5, 8, 8, 7, 5, 12, 6, 8, 7, 10, 6, 14, 7, 10, 10, 10, 10, 14, 8, 11, 12, 16, 8, 18, 9, 14, 14, 13, 9, 20, 11, 16, 14, 16, 10, 19, 14, 20, 16, 16, 11, 28, 12, 17, 18, 18, 16, 26, 13, 20, 18, 26, 13, 28

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

Examples

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

References

• 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 = 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
• Index entries for sequences related to A2 = hexagonal = triangular lattice
• Index entries for sequences related to sublattices

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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).