

A145394


Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/3 to give the other.


10



1, 1, 2, 3, 2, 4, 4, 5, 5, 6, 4, 10, 6, 8, 8, 11, 6, 13, 8, 14, 12, 12, 8, 20, 11, 14, 14, 20, 10, 24, 12, 21, 16, 18, 16, 31, 14, 20, 20, 30, 14, 32, 16, 28, 26, 24, 16, 42, 21, 31, 24, 34, 18, 40, 24, 40, 28, 30, 20, 56, 22, 32, 36, 43, 28, 48, 24, 42, 32, 48, 24, 65, 26, 38, 42, 48, 32, 56, 28, 62
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OFFSET

1,3


COMMENTS

Also, apparently a(n) is the number of nonequivalent (up to latticepreserving affine transformation) triangles on 2D square lattice of area n/2 [Karpenkov].  Andrey Zabolotskiy, Jul 06 2017
From Andrey Zabolotskiy, Jan 18 2018: (Start)
The parent lattice of the sublattices under consideration has Patterson symmetry group p6, 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), A003051 (p6mm).
If we count sublattices related by parentlatticepreserving reflection as equivalent, we get A003051 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/3; see illustration in links) as equivalent, we get A054384. If we count sublattices related by any rotation or reflection as equivalent, we get A300651.
Rutherford says at p. 161 that a(n) != A054384(n) only when A002324(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 14 (see illustration). (End)


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384
Oleg Karpenkov, Elementary notions of lattice trigonometry, Mathematica Scandinavica, vol.102, no.2, pp.161205, (2008) [See page 203].
Oleg Karpenkov, Geometry of Lattice Angles, Polygons, and Cones, Thesis, Technische Universität Graz, 2009.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the numbertheoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1610.07793 [math.NT], 2016.
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) = (A000203(n) + 2 * A002324(n))/3. [Rutherford]  N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2n } A000086(n/m^2) + A157227(n/m^2) = A002324(n) + Sum_{ m: m^2n } A157227(n/m^2). [Rutherford]  Andrey Zabolotskiy, Apr 23 2018
a(n) = Sum_{ dn } A008611(d1).  Andrey Zabolotskiy, Aug 29 2019


MATHEMATICA

a[n_] := (DivisorSigma[1, n] + 2 DivisorSum[n, Switch[Mod[#, 3], 1, 1, 2, 1, 0, 0] &])/3; Array[a, 80] (* JeanFrançois Alcover, Dec 03 2015 *)


PROG

(PARI)
A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1)  (d%3==2)));
A000203(n) = if( n<1, 0, sigma(n));
a(n) = (A000203(n) + 2 * A002324(n)) / 3;
\\ Joerg Arndt, Oct 13 2013


CROSSREFS

Cf. A054384, A000203, A069734, A145391, A145392, A145393, A003051, A002324, A002654, A069735, A145390, A300651, A000086, A157227, A008611.
Sequence in context: A324750 A320348 A336315 * A179806 A182762 A173997
Adjacent sequences: A145391 A145392 A145393 * A145395 A145396 A145397


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Feb 23 2009


EXTENSIONS

New name from Andrey Zabolotskiy, Dec 14 2017


STATUS

approved



