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%I #34 Jul 18 2019 01:14:01
%S 1,2,2,4,3,5,3,7,5,7,4,11,5,8,8,12,6,13,6,15,10,11,7,21,10,13,12,18,9,
%T 22,9,21,14,16,14,29,11,17,16,29,12,28,12,25,23,20,13,39,16,27,20,29,
%U 15,34,20,36,22,25,16,50,17,26,29,38,24,40,18,36,26,40
%N Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.
%C From _Andrey Zabolotskiy_, Mar 12 2018: (Start)
%C If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.
%C The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, 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), A145394 (p6), A003051 (p6mm).
%C Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)
%H Andrey Zabolotskiy, <a href="/A145393/b145393.txt">Table of n, a(n) for n = 1..10000</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 6 and fig. 2).
%H 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; beware the typo in a(5).]
%H Andrey Zabolotskiy, <a href="/A145392/a145392.pdf">Sublattices of the square lattice</a> (illustrations for n = 1..6, 15, 25)
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H <a href="/index/Sq#sqlatt">Index entries for sequences related to square lattice</a>
%F a(n) = (A000203(n) + A002654(n) + A069735(n) + A145390(n))/4. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F G.f.: Sum_{ m>=1 } (1/((1-x^m)(1-x^(4m))) - 1). [Hanany, Orlando & Reffert, eq. (6.8)] - _Andrey Zabolotskiy_, Jul 05 2017
%F a(n) = Sum_{ m: m^2|n } A019590(n/m^2) + A157228(n/m^2) + A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2) = A053866(n) + A025441(n) + Sum_{ m: m^2|n } A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2). [Rutherford] - _Andrey Zabolotskiy_, May 07 2018
%F a(n) = Sum_{ d|n } A008621(d) = Sum_{ d|n } (1 + floor(d/4)). [From the above-given g.f.] - _Andrey Zabolotskiy_, Jul 17 2019
%t terms = 70;
%t CoefficientList[Sum[(1/((1-x^m)(1-x^(4m)))-1), {m, 1, terms}] + O[x]^(terms + 1), x] // Rest (* _Jean-François Alcover_, Aug 05 2018 *)
%Y Cf. A054345, A054346, A167156, A008621.
%Y Cf. A000203, A069734, A145391, A145392, A145394, A003051, A002324, A002654, A069735, A145390.
%Y Cf. A019590, A157228, A157226, A157230, A157231, A053866, A025441.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 23 2009
%E New name from _Andrey Zabolotskiy_, Mar 12 2018
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