

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)


8



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
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OFFSET

1,3


COMMENTS

The hexagonal lattice is the familiar 2dimensional 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 halfintegral area is infinite, the triangles are probably counted up to the latticepreserving affine transformation.  Andrey Zabolotskiy, Mar 10 2018]
Also number of 2nvertex connected cubic vertextransitive graphs which are Cayley graphs for a dihedral group [Potočnik et al.].  N. J. A. Sloane, Apr 19 2014


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), 201217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201217. [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) 2939 (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 vertextransitive graphs (See the subpage 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^(w2+C_2),
where C_1 = 0 if 2n or 9n, = 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 (not only primitive sublattices), A001615, A006984, A007997, A048259, A054345.
Sequence in context: A078342 A177903 A107325 * A070868 A272612 A155216
Adjacent sequences: A003047 A003048 A003049 * A003051 A003052 A003053


KEYWORD

nonn,nice,changed


AUTHOR

N. J. A. Sloane


STATUS

approved



