A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.

1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 6, 4, 10, 5, 7, 8, 11, 6, 13, 7, 14, 10, 12, 8, 20, 11, 13, 14, 17, 10, 24, 11, 21, 16, 18, 14, 31, 13, 19, 18, 30, 14, 28, 15, 28, 26, 24, 16, 42, 17, 31, 24, 31, 18, 40, 24, 35, 26, 30, 20, 56, 21, 31, 31, 43, 26, 48, 23, 42, 32, 42, 24, 65

Offset

0,4

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

If reflections are allowed, we get A300651. If only rotations that preserve the parent hexagonal lattice are allowed, we get A145394. The analog for square lattice is A054345. - Andrey Zabolotskiy, Mar 10 2018

Links

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

M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).

Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^SL_{x^2+xy+y^2}(s).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

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 [see Table 2]. - From N. J. A. Sloane, Feb 23 2009

Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)

Index entries for sequences related to sublattices

Index entries for sequences related to A2 = hexagonal = triangular lattice

Prog

(SageMath)
# see A159842 for the definitions of dc, fin, u, N
def gg(m, k1, minus = True):
    def f(n):
        if n == 1: return 1
        r = 1
        for (p, k) in factor(n):
            if p % 3 != m or k1 and k > 1: return 0
            if minus: r *= (-1)**k
        return r
    return f
g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False)
def a_SL(n):
    return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3
print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

Crossrefs

Cf. A003051, A054345, A054346, A145394, A300651.

Sequence in context: A283368 A199474 A246694 * A026400 A026409 A336215

Adjacent sequences: A054381 A054382 A054383 * A054385 A054386 A054387

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, May 08 2000

Status

approved