

A003051


Number of inequivalent sublattices of index n in hexagonal lattice (two sublattices are equivalent if one can be rotated or reflected to give the other).
(Formerly M0420)


24



1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 18, 9, 17, 16, 13, 9, 28, 12, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 21, 27, 16, 26, 13, 24, 18, 26, 13, 40, 14
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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.
From Andrey Zabolotskiy, Mar 10 2018: (Start)
If only primitive sublattices are considered, we get A003050.
Here only rotations and reflections preserving the parent hexagonal lattice are allowed. If reflections are not allowed, we get A145394. If any rotations and reflections are allowed, we get A300651.
In other words, the parent lattice of the sublattices under consideration has Patterson symmetry group p6mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6).
Rutherford says at p. 161 that his sequence for p6mm differs from this sequence, but it seems that with the current definition and terms of this sequence, this actually is his p6mm sequence, and the sequence he thought to be this one is actually A300651. Also, he says that a(n) != A300651(n) only when A002324(n) > 2 (first time happens at n = 49), but actually these two sequences differ at other terms, too, for example, at n = 42 (see illustration). (End)


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Vincenzo Librandi, 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).
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hepth] (see Table 3).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156163. [See Table 2].
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice


FORMULA

a(n) = Sum_{ m^2  n } A003050(n/m^2).
a(n) = (A000203(n) + 2*A002324(n) + 3*A145390(n))/6. [Rutherford]  N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ dn } A112689(d+1).  Andrey Zabolotskiy, Aug 29 2019


MATHEMATICA

max = 73; A145390 = Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1  x^n), {n, 1, max}], {x, 0, max}], x], 1]; A002324[n_] := (dn = Divisors[n]; Count[dn, _?(Mod[#, 3] == 1 & )]  Count[dn, _?(Mod[#, 3] == 2 & )]); a[n_] := (DivisorSigma[1, n] + 2 A002324[n] + 3*A145390[[n]])/6; Table[a[n], {n, 1, max}] (* JeanFrançois Alcover, Oct 11 2011, after given formula *)


CROSSREFS

Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.
Sequence in context: A036762 A032154 A300651 * A305866 A328406 A257396
Adjacent sequences: A003048 A003049 A003050 * A003052 A003053 A003054


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



