A300651 - OEIS

%I #24 Sep 27 2024 05:43:01

%S 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,

%T 10,10,10,20,8,11,12,20,8,17,9,17,16,13,9,28,11,17,14,20,10,22,14,25,

%U 16,16,11,34,12,17,20,27,16,26,13,24,18,24,13,40

%N 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.

%C 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.

%C 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.

%H Andrey Zabolotskiy, <a href="/A300651/b300651.txt">Table of n, a(n) for n = 1..1000</a>

%H Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, <a href="https://arxiv.org/abs/2409.05625">Zeta functions enumerating subforms of quadratic forms</a>, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^GL_{x^2+xy+y^2}(s).

%H Andrey Zabolotskiy, <a href="/A145394/a145394.pdf">Sublattices of the hexagonal lattice</a> (illustrations for n = 1..7, 14)

%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>

%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>

%o (SageMath)

%o # See A159842 and A054384 for the definitions of functions used here

%o def a_GL(n):

%o     return (a_SL(n) + dc(fin(1, -1, 0, 2), u, u, g2)(n)) / 2

%o print([a_GL(n) for n in range(1, 100)]) # _Andrey Zabolotskiy_, Sep 22 2024

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

%K nonn

%O 1,3

%A _Andrey Zabolotskiy_, Mar 10 2018