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 A145391 Number of sublattices of index n of a 2D centered rectangular lattice fixed by the c2mm (cmm) Patterson symmetry operations of the parent lattice. 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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.] 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) [conjecture]. - 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 Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525. Sequence in context: A085180 A114750 A234923 * A057756 A160051 A082746 Adjacent sequences:  A145388 A145389 A145390 * A145392 A145393 A145394 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 23 2009 EXTENSIONS New name from Andrey Zabolotskiy, Mar 12 2018 STATUS approved

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Last modified August 8 06:27 EDT 2020. Contains 336290 sequences. (Running on oeis4.)