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.

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

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