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%I M0420 #82 Feb 24 2023 12:29:07
%S 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,
%T 10,10,10,20,8,11,12,20,8,18,9,17,16,13,9,28,12,17,14,20,10,22,14,25,
%U 16,16,11,34,12,17,21,27,16,26,13,24,18,26,13,40,14
%N Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.
%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 From _Andrey Zabolotskiy_, Mar 10 2018: (Start)
%C If only primitive sublattices are considered, we get A003050.
%C 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.
%C 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).
%C 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)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003051/b003051.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, <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 Amihay Hanany, Domenico Orlando, and Susanne Reffert, <a href="https://doi.org/10.1007/JHEP06(2010)051">Sublattice counting and orbifolds</a>, High Energ. Phys., 2010 (2010), 51, <a href="https://arxiv.org/abs/1002.2981">arXiv.org:1002.2981 [hep-th]</a> (see Table 3).
%H W. Kurth, <a href="https://doi.org/10.1016/0012-365X(86)90029-4">Enumeration of Platonic maps on the torus</a>, Discrete Math. 61 (1986), 71-83.
%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="https://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].
%H Andrey Zabolotskiy, <a href="/A145394/a145394.pdf">Sublattices of the hexagonal lattice</a> (illustrations for n = 1..7, 14)
%H Andrey Zabolotskiy, <a href="https://arxiv.org/abs/2003.10251">Coweight lattice A^*_n and lattice simplices</a>, arXiv:2003.10251 [math.CO], 2020.
%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>
%F a(n) = Sum_{ m^2 | n } A003050(n/m^2).
%F a(n) = (A000203(n) + 2*A002324(n) + 3*A145390(n))/6. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F a(n) = Sum_{ d|n } A112689(d+1). - _Andrey Zabolotskiy_, Aug 29 2019
%F a(n) = Sum_{ d|n } floor(d/6) + 1 - 1*[d == 2 or 6 (mod 12)] + 1*[d == 4 (mod 12)]. [Kurth] - _Brahadeesh Sankarnarayanan_, Feb 24 2023
%t 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}] (* _Jean-François Alcover_, Oct 11 2011, after given formula *)
%Y Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.
%K nonn,nice,easy
%O 1,3
%A _N. J. A. Sloane_
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