A145391 - OEIS

A145391

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

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, 22, 31, 16, 38, 17, 36, 26, 28, 26, 50, 20, 31, 30, 50, 22, 50, 23, 45, 42, 37, 25, 69, 30, 48, 38, 52, 28, 62, 38, 65, 42, 46, 31, 90, 32, 49, 55, 69, 44, 74, 35, 66, 50, 74

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OFFSET

1,2

COMMENTS

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

LINKS

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000

Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] [see Table 8].

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2.]

Index entries for sequences related to sublattices

FORMULA

a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009

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

a(n) = Sum_{ d|n } A004525(d+1). - Andrey Zabolotskiy, Aug 29 2019

MATHEMATICA

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)];

a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];

a[n_] := (DivisorSigma[1, n] + a145390[n])/2;

Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

CROSSREFS