A054345 - OEIS

%I #33 Sep 27 2024 05:43:13

%S 1,1,2,2,4,3,6,4,8,7,8,6,14,7,12,10,16,9,20,10,18,16,18,12,30,13,20,

%T 20,28,15,30,16,32,24,26,20,46,19,30,26,38,21,48,22,42,33,36,24,62,29,

%U 38,34,46,27,60,30,60,40,44,30,70,31,48,52,64,33,72,34,60,48,60

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

%C If reflections are allowed, we get A054346. If only rotations that preserve the parent square lattice are allowed, we get A145392. The analog for a hexagonal lattice is A054384.

%H Andrey Zabolotskiy, <a href="/A054345/b054345.txt">Table of n, a(n) for n = 0..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 6.2 for the Dirichlet g.f. zeta^SL_{x^2+y^2}(s).

%H 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 Andrey Zabolotskiy, <a href="/A145392/a145392.pdf">Sublattices of the square lattice</a> (illustrations for n = 1..6, 15, 25).

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

%H <a href="/index/Sq#sqlatt">Index entries for sequences related to square lattice</a>

%e For n = 1, 2, 3, 4 the sublattices are generated by the rows of:

%e   [1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]

%e   [0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].

%o (SageMath)

%o # see A159842 for the definitions of dc, fin, u, N

%o def ff(m, k1, minus = True):

%o     def f(n):

%o         if n == 1: return 1

%o         r = 1

%o         for (p, k) in factor(n):

%o             if p % 4 != m or k1 and k > 1: return 0

%o             if minus: r *= (-1)**k

%o         return r

%o     return f

%o f1, f2, f3 = ff(1, True), ff(1, True, False), ff(3, False)

%o def a_SL(n):

%o     return (dc(u, N, f1)(n) + dc(u, f3)(n)) / 2

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

%Y Cf. A003051, A001615, A054346, A054384, A145392.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, May 06 2000