A054346 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.

1, 1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 9, 13, 12, 18, 9, 21, 9, 21, 14, 16, 13, 29, 11, 17, 16, 28, 12, 28, 12, 25, 21, 20, 13, 39, 16, 24, 20, 29, 15, 34, 18, 36, 22, 25, 16, 47, 17, 26, 29, 38, 21, 40, 18, 36, 26, 36, 19, 58, 20

Offset

0,3

Comments

If we count sublattices as equivalent only if they are related by a rotation, we get A054345 instead of this sequence. If we only allow rotations and reflections that preserve the parent (square) lattice, we get A145393; the first discrepancy is at n = 25 (see illustration), the second is at n = 30. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145392. The analog for the hexagonal lattice is A300651. - Andrey Zabolotskiy, Mar 12 2018

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000

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. - From N. J. A. Sloane, Feb 23 2009

Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)

Index entries for sequences related to sublattices

Index entries for sequences related to square lattice

Example

For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
  [1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
  [0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].

Crossrefs

Cf. A003051, A001615, A054345.

Sequence in context: A133438 A086671 A269502 * A145393 A215675 A329439

Adjacent sequences: A054343 A054344 A054345 * A054347 A054348 A054349

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, May 06 2000

Status

approved