A300651 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if they are related by any rotation or reflection.

1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 17, 9, 17, 16, 13, 9, 28, 11, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 20, 27, 16, 26, 13, 24, 18, 24, 13, 40

Offset

1,3

Comments

If we count sublattices as equivalent only if they are related by a rotation, we get A054384 instead of this sequence. If we only allow rotations and reflections that preserve the parent (hexagonal) lattice, we get A003051; the first discrepancy is at n = 42 (see illustration), the second is at n = 49. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145394. The analog for square lattice is A054346.

Although A003051 has its counterpart A003050 which counts primitive sublattices only, this sequence has no such counterpart sequence because a primitive sublattice can turn to a non-primitive one via a non-parent-lattice-preserving rotation, so the straightforward definition of primitiveness does not work in this case.

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000

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

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 and A054384 for the definitions of functions used here
def a_GL(n):
    return (a_SL(n) + dc(fin(1, -1, 0, 2), u, u, g2)(n)) / 2
print([a_GL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

Crossrefs

Cf. A003050, A003051, A054384, A145394, A054346.

Sequence in context: A217438 A036762 A032154 * A003051 A377774 A305866

Adjacent sequences: A300648 A300649 A300650 * A300652 A300653 A300654

Keyword

nonn

Author

Andrey Zabolotskiy, Mar 10 2018

Status

approved