A054345 Number of inequivalent sublattices of index n in a square lattice, where two sublattices are considered equivalent if one can be rotated to give the other.

1, 1, 2, 2, 4, 3, 6, 4, 8, 7, 8, 6, 14, 7, 12, 10, 16, 9, 20, 10, 18, 16, 18, 12, 30, 13, 20, 20, 28, 15, 30, 16, 32, 24, 26, 20, 46, 19, 30, 26, 38, 21, 48, 22, 42, 33, 36, 24, 62, 29, 38, 34, 46, 27, 60, 30, 60, 40, 44, 30, 70, 31, 48, 52, 64, 33, 72, 34, 60, 48, 60

Offset

0,3

Comments

If reflections are allowed, we get A054346. If only rotations that preserve the parent square lattice are allowed, we get A145392. The analog for a hexagonal lattice is A054384.

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^SL_{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

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].

Prog

(SageMath)
# see A159842 for the definitions of dc, fin, u, N
def ff(m, k1, minus = True):
    def f(n):
        if n == 1: return 1
        r = 1
        for (p, k) in factor(n):
            if p % 4 != m or k1 and k > 1: return 0
            if minus: r *= (-1)**k
        return r
    return f
f1, f2, f3 = ff(1, True), ff(1, True, False), ff(3, False)
def a_SL(n):
    return (dc(u, N, f1)(n) + dc(u, f3)(n)) / 2
print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

Crossrefs

Cf. A003051, A001615, A054346, A054384, A145392.

Sequence in context: A365195 A365433 A029578 * A368582 A352956 A304214

Adjacent sequences: A054342 A054343 A054344 * A054346 A054347 A054348

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, May 06 2000

Status

approved