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%I #26 Aug 31 2023 10:02:35
%S 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 9,21,9,21,14,16,13,29,11,17,16,28,12,28,12,25,21,20,13,39,16,24,20,
%U 29,15,34,18,36,22,25,16,47,17,26,29,38,21,40,18,36,26,36,19,58,20
%N 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 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 Andrey Zabolotskiy, <a href="/A054346/b054346.txt">Table of n, a(n) for n = 0..1000</a>
%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].
%Y Cf. A003051, A001615, A054345.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, May 06 2000
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