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%I #38 Jan 19 2022 14:31:02
%S 1,2,3,5,4,7,5,10,8,10,7,17,8,13,14,19,10,21,11,24,18,19,13,35,17,22,
%T 22,31,16,38,17,36,26,28,26,50,20,31,30,50,22,50,23,45,42,37,25,69,30,
%U 48,38,52,28,62,38,65,42,46,31,90,32,49,55,69,44,74,35,66,50,74
%N Number of inequivalent sublattices of index n in centered rectangular lattice.
%C The centered rectangular lattice has symmetry group c2mm, or cmm. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - _Andrey Zabolotskiy_, Mar 12 2018
%H Andrey Zabolotskiy, <a href="/A145391/b145391.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 8].
%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 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%F a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F a(n) = Sum_{ m: m^2|n } A060594(n/m^2) + A157223(n/m^2) = A145390(n) + Sum_{ m: m^2|n } A157223(n/m^2). - _Andrey Zabolotskiy_, May 07 2018
%F a(n) = Sum_{ d|n } A004525(d+1). - _Andrey Zabolotskiy_, Aug 29 2019
%t a060594[n_] := Switch[Mod[n, 8], 2|6, 2^(PrimeNu[n] - 1), 1|3|4|5|7, 2^PrimeNu[n], 0, 2^(PrimeNu[n] + 1)];
%t a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];
%t a[n_] := (DivisorSigma[1, n] + a145390[n])/2;
%t Array[a, 100] (* _Jean-François Alcover_, Aug 31 2018 *)
%Y Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 23 2009
%E New name from _Andrey Zabolotskiy_, Mar 12 2018
%E New name from _Andrey Zabolotskiy_, Jan 19 2022
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