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A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

%I A145391 #38 Jan 19 2022 14:31:02
%S A145391 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 A145391 22,31,16,38,17,36,26,28,26,50,20,31,30,50,22,50,23,45,42,37,25,69,30,
%U A145391 48,38,52,28,62,38,65,42,46,31,90,32,49,55,69,44,74,35,66,50,74
%N A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.
%C A145391 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 A145391 Andrey Zabolotskiy, <a href="/A145391/b145391.txt">Table of n, a(n) for n = 1..10000</a>
%H A145391 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 A145391 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 A145391 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%F A145391 a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F A145391 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 A145391 a(n) = Sum_{ d|n } A004525(d+1). - _Andrey Zabolotskiy_, Aug 29 2019
%t A145391 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 A145391 a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];
%t A145391 a[n_] := (DivisorSigma[1, n] + a145390[n])/2;
%t A145391 Array[a, 100] (* _Jean-François Alcover_, Aug 31 2018 *)
%Y A145391 Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525.
%K A145391 nonn
%O A145391 1,2
%A A145391 _N. J. A. Sloane_, Feb 23 2009
%E A145391 New name from _Andrey Zabolotskiy_, Mar 12 2018
%E A145391 New name from _Andrey Zabolotskiy_, Jan 19 2022