cp's OEIS Frontend

A054346 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.

%I A054346 #35 Sep 27 2024 05:43:26
%S A054346 1,1,2,2,4,3,5,3,7,5,7,4,11,5,8,8,12,6,13,6,15,10,11,7,21,9,13,12,18,
%T A054346 9,21,9,21,14,16,13,29,11,17,16,28,12,28,12,25,21,20,13,39,16,24,20,
%U A054346 29,15,34,18,36,22,25,16,47,17,26,29,38,21,40,18,36,26,36,19,58,20
%N A054346 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.
%C A054346 If we count sublattices as equivalent only if they are related by a rotation, we get A054345 instead of this sequence. If we only allow rotations and reflections that preserve the parent (square) lattice, we get A145393; the first discrepancy is at n = 25 (see illustration), the second is at n = 30. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145392. The analog for the hexagonal lattice is A300651. - _Andrey Zabolotskiy_, Mar 12 2018
%H A054346 Andrey Zabolotskiy, <a href="/A054346/b054346.txt">Table of n, a(n) for n = 0..1000</a>
%H A054346 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 6.2 for the Dirichlet g.f. zeta^GL_{x^2+y^2}(s).
%H A054346 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. - From _N. J. A. Sloane_, Feb 23 2009
%H A054346 Andrey Zabolotskiy, <a href="/A145392/a145392.pdf">Sublattices of the square lattice</a> (illustrations for n = 1..6, 15, 25)
%H A054346 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H A054346 <a href="/index/Sq#sqlatt">Index entries for sequences related to square lattice</a>
%e A054346 For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
%e A054346   [1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
%e A054346   [0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].
%o A054346 (SageMath)
%o A054346 # See A159842 and A054345 for the definitions of functions used here
%o A054346 def a_GL(n):
%o A054346     return (a_SL(n) + dc(fin(1, 0, 0, 1), u, u, f2)(n)) / 2
%o A054346 print([a_GL(n) for n in range(1, 100)]) # _Andrey Zabolotskiy_, Sep 22 2024
%Y A054346 Cf. A003051, A001615, A054345.
%K A054346 nonn,easy,nice
%O A054346 0,3
%A A054346 _N. J. A. Sloane_, May 06 2000