cp's OEIS Frontend

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.

%I A054384 #36 Sep 27 2024 05:43:37
%S A054384 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,
%T A054384 10,24,11,21,16,18,14,31,13,19,18,30,14,28,15,28,26,24,16,42,17,31,24,
%U A054384 31,18,40,24,35,26,30,20,56,21,31,31,43,26,48,23,42,32,42,24,65
%N 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.
%C A054384 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A054384 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
%H A054384 Andrey Zabolotskiy, <a href="/A054384/b054384.txt">Table of n, a(n) for n = 0..1000</a>
%H A054384 M. Bernstein, N. J. A. Sloane and P. E. Wright, <a href="https://doi.org/10.1016/0012-365X(95)00354-Y">On Sublattices of the Hexagonal Lattice</a>, Discrete Math. 170 (1997) 29-39 (<a href="http://neilsloane.com/doc/paul.txt">Abstract</a>, <a href="http://neilsloane.com/doc/paul.pdf">pdf</a>, <a href="http://neilsloane.com/doc/paul.ps">ps</a>).
%H A054384 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^SL_{x^2+xy+y^2}(s).
%H A054384 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H A054384 John S. Rutherford, <a href="https://doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163 [see Table 2]. - From _N. J. A. Sloane_, Feb 23 2009
%H A054384 Andrey Zabolotskiy, <a href="/A145394/a145394.pdf">Sublattices of the hexagonal lattice</a> (illustrations for n = 1..7, 14)
%H A054384 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H A054384 <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%o A054384 (SageMath)
%o A054384 # see A159842 for the definitions of dc, fin, u, N
%o A054384 def gg(m, k1, minus = True):
%o A054384     def f(n):
%o A054384         if n == 1: return 1
%o A054384         r = 1
%o A054384         for (p, k) in factor(n):
%o A054384             if p % 3 != m or k1 and k > 1: return 0
%o A054384             if minus: r *= (-1)**k
%o A054384         return r
%o A054384     return f
%o A054384 g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False)
%o A054384 def a_SL(n):
%o A054384     return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3
%o A054384 print([a_SL(n) for n in range(1, 100)]) # _Andrey Zabolotskiy_, Sep 22 2024
%Y A054384 Cf. A003051, A054345, A054346, A145394, A300651.
%K A054384 nonn,nice,easy
%O A054384 0,4
%A A054384 _N. J. A. Sloane_, May 08 2000