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A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two lattices are considered equivalent if one can be rotated to give the other. 5

%I

%S 1,1,1,2,3,2,4,3,5,5,6,4,10,5,7,8,11,6,13,7,14,10,12,8,20,11,13,14,17,

%T 10,24,11,21,16,18,14,31,13,19,18,30,14,28,15,28,26,24,16,42,17,31,24,

%U 31,18,40,24,35,26,30,20,56,21,31,31,43,26,48,23,42,32,42,24,65

%N Number of inequivalent sublattices of index n in hexagonal lattice, where two lattices are considered equivalent if one can be rotated to give the other.

%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 If reflections are allowed, we get A300651. If only rotations that preserve the parent hexagonal lattice are allowed, we get A145394. The analog for square lattice is A054345. - _Andrey Zabolotskiy_, Mar 10 2018

%H Andrey Zabolotskiy, <a href="/A054384/b054384.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Bernstein, N. J. A. Sloane and P. E. Wright, <a href="https://doi.org/10.1016/0012-365X(95)00354-Y">On Sublattices of the Hexagonal Lattice</a>, 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 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 John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163 [see Table 2]. - From _N. J. A. Sloane_, Feb 23 2009

%H Andrey Zabolotskiy, <a href="/A145394/a145394.pdf">Sublattices of the hexagonal lattice</a> (illustrations for n = 1..7, 14)

%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>

%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>

%Y Cf. A003051, A054345, A054346, A145394, A300651.

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_, May 08 2000

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Last modified July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)