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%I M0229 #63 Sep 19 2023 08:20:25
%S 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,
%T 10,10,10,14,8,11,12,16,8,18,9,14,14,13,9,20,11,16,14,16,10,19,14,20,
%U 16,16,11,28,12,17,18,18,16,26,13,20,18,26,13,28
%N 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.
%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C 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]
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003050/b003050.txt">Table of n, a(n) for n = 1..1000</a>
%H A. Altshuler, <a href="https://doi.org/10.1016/S0012-365X(73)80002-0">Construction and enumeration of regular maps on the torus</a>, Discrete Math. 4 (1973), 201-217.
%H A. Altshuler, <a href="/A003050/a003050.pdf">Construction and enumeration of regular maps on the torus</a>, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
%H M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (<a href="http://neilsloane.com/doc/paul.txt">Abstract</a>, <a href="http://neilsloane.com/doc/paul.pdf">pdf</a>, <a href="http://neilsloane.com/doc/paul.ps">ps</a>).
%H J. Davey, A. Hanany and Rak-Kyeong Seong, <a href="http://arxiv.org/abs/1002.3609">Counting Orbifolds</a>, arXiv:1002.3609 [hep-th], 2010.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.
%H Primož Potočnik, Pablo Spiga and Gabriel Verret, <a href="http://staff.matapp.unimib.it/~spiga/census.html">A census of small connected cubic vertex-transitive graphs</a> (See the sub-page Table.html, column headed "Dihedrants"). - _N. J. A. Sloane_, Apr 19 2014
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%F 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),
%F where C_1 = 0 if 2|n or 9|n, = Product_{i=1..w, p_i > 3} (1 + Legendre(p_i, 3)) otherwise,
%F 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).
%e For n = 6 the 3 primitive triples are 510, 411, 321.
%t 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 *)
%Y Cf. A003051 (not only primitive sublattices), A001615, A006984, A007997, A048259, A054345, A154272, A157235.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_
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