This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A145394 #67 Jul 03 2020 06:22:04
%S A145394 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,
%T A145394 10,24,12,21,16,18,16,31,14,20,20,30,14,32,16,28,26,24,16,42,21,31,24,
%U A145394 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
%N 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.
%C A145394 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
%C A145394 From _Andrey Zabolotskiy_, Jan 18 2018: (Start)
%C A145394 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).
%C A145394 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.
%C A145394 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)
%H A145394 Antti Karttunen, <a href="/A145394/b145394.txt">Table of n, a(n) for n = 1..16384</a>
%H A145394 Oleg Karpenkov, <a href="http://www.mscand.dk/article/viewFile/15058/13053">Elementary notions of lattice trigonometry</a>, Mathematica Scandinavica, vol.102, no.2, pp.161-205, (2008) [See page 203].
%H A145394 Oleg Karpenkov, <a href="http://pcwww.liv.ac.uk/~karpenk/hab/habilitation.pdf">Geometry of Lattice Angles, Polygons, and Cones</a>, Thesis, Technische Universität Graz, 2009.
%H A145394 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, 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.]
%H A145394 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.
%H A145394 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.]
%H A145394 Andrey Zabolotskiy, <a href="/A145394/a145394.pdf">Sublattices of the hexagonal lattice</a> (illustrations for n = 1..7, 14)
%H A145394 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 A145394 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H A145394 <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A145394 a(n) = (A000203(n) + 2 * A002324(n))/3. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F A145394 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
%F A145394 a(n) = Sum_{ d|n } A008611(d-1). - _Andrey Zabolotskiy_, Aug 29 2019
%t A145394 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 *)
%o A145394 (PARI)
%o A145394 A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)));
%o A145394 A000203(n) = if( n<1, 0, sigma(n));
%o A145394 a(n) = (A000203(n) + 2 * A002324(n)) / 3;
%o A145394 \\ _Joerg Arndt_, Oct 13 2013
%Y A145394 Cf. A054384, A000203, A069734, A145391, A145392, A145393, A003051, A002324, A002654, A069735, A145390, A300651, A000086, A157227, A008611.
%K A145394 nonn
%O A145394 1,3
%A A145394 _N. J. A. Sloane_, Feb 23 2009
%E A145394 New name from _Andrey Zabolotskiy_, Dec 14 2017