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A054346 Number of inequivalent sublattices of index n in square lattice, where two lattices are considered equivalent if one can be rotated or reflected to give the other. 7
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 (list; graph; refs; listen; history; text; internal format)
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 A132802

Adjacent sequences:  A054343 A054344 A054345 * A054347 A054348 A054349

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, May 06 2000

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)