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 A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n. 7
 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1, 4, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS After a(2), this matches A034380 except for n = 63, 65, 80, 85, ... - R. J. Mathar, Feb 27 2009 [Updated by Andrey Zabolotskiy, May 09 2018] LINKS Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000 J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Act. Cryst. A65 (2009) 156-163, Table 5 symmetry *mm2. FORMULA From Andrey Zabolotskiy, Sep 30 2018: (Start) a(n) = (A060594(n) - A019590(n))/2. a(n) = 2^(A046072(n)-1) for n>2. Thus a(n) = 1 if n>2 is in A033948, a(n) = 2 if n is in A272592, etc. (End) MATHEMATICA a[n_] := If[n <= 2, 0, Sum[Boole[Mod[k^2, n] == 1], {k, 1, n}]/2]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 12 2023 *) CROSSREFS Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157231, A304182, A060594, A046072, A033948, A272592. Sequence in context: A332761 A046072 A072273 * A034380 A328966 A077479 Adjacent sequences: A157227 A157228 A157229 * A157231 A157232 A157233 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 25 2009 EXTENSIONS New name and more terms from Andrey Zabolotskiy, May 09 2018 STATUS approved

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Last modified February 28 22:27 EST 2024. Contains 370400 sequences. (Running on oeis4.)