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

Author

N. J. A. Sloane, May 06 2000

Keywords

• nonn
• easy
• nice

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

Examples

			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].
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
• Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.2 for the Dirichlet g.f. zeta^GL_{x^2+y^2}(s).
• 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

Crossrefs

Cf. A003051, A001615, A054345.

Programs

SageMath

# See A159842 and A054345 for the definitions of functions used here
def a_GL(n):
    return (a_SL(n) + dc(fin(1, 0, 0, 1), u, u, f2)(n)) / 2
print([a_GL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024