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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.
(Formerly M0229)
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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

REFERENCES

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

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

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).

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

FORMULA

Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6) * n prod_{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, = prod_{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.

EXAMPLE

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

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 *)

CROSSREFS

Cf. A003051, A001615, A006984, A007997, A048259, A054345.

Sequence in context: A078342 A177903 A107325 * A070868 A155216 A064144

Adjacent sequences:  A003047 A003048 A003049 * A003051 A003052 A003053

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 27 17:19 EDT 2015. Contains 261095 sequences.