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.

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

Offset: 1

Author

N. J. A. Sloane, Feb 23 2009

Keywords

• nonn

Comments

Also, apparently a(n) is the number of nonequivalent (up to lattice-preserving 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 parent-lattice-preserving 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.161-205, (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 two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical 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 two-dimensional 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, 156-163. [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

Crossrefs

Cf. A054384, A000203, A069734, A145391, A145392, A145393, A003051, A002324, A002654, A069735, A145390, A300651, A000086, A157227, A008611.

Programs

Mathematica

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

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

Formula

a(n) = (A000203(n) + 2 * A002324(n))/3. [Rutherford] - N. J. A. Sloane, Mar 13 2009

a(n) = Sum_{ m: m^2|n } A000086(n/m^2) + A157227(n/m^2) = A002324(n) + Sum_{ m: m^2|n } A157227(n/m^2). [Rutherford] - Andrey Zabolotskiy, Apr 23 2018

a(n) = Sum_{ d|n } A008611(d-1). - Andrey Zabolotskiy, Aug 29 2019

Extensions

New name from Andrey Zabolotskiy, Dec 14 2017