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)
PROG
(SageMath)
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
KEYWORD
nonn
AUTHOR
Andrey Zabolotskiy, Mar 10 2018
STATUS
approved