%I A003051 M0420
%S A003051 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,
%T A003051 8,9,13,6,14,7,15,10,10,10,20,8,11,12,20,8,18,9,17,16,13,9,28,12,
%U A003051 17,14,20,10,22,14,25,16,16,11,34,12,17,21,27,16,26,13,24,18,26,13,40,
               14
%N A003051 Number of inequivalent sublattices of index n in hexagonal lattice (two 
               sublattices are equivalent if one can be rotated or reflected to 
               give the other).
%C A003051 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%D A003051 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003051 A. Altshuler, Construction and enumeration of regular maps on the torus, 
               Discrete Math. 4 (1973), 201-217.
%D A003051 John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of 
               plane sublattices by parent Patterson symmetry and colour lattice 
               group type, Acta Cryst. (2009). A65, 156163. [See Table 2]. [From 
               N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
%H A003051 M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the 
               Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (<a href="http:/
               /www.research.att.com/~njas/doc/paul.txt">Abstract</a>, <a href="http:/
               /www.research.att.com/~njas/doc/paul.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/paul.ps">ps</a>).
%H A003051 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/A2.html">Home page for hexagonal (or triangular) lattice 
               A2</a>
%H A003051 <a href="Sindx_Su.html#sublatts">Index entries for sequences related 
               to sublattices</a>
%H A003051 <a href="Sindx_Aa.html#A2">Index entries for sequences related to A2 
               = hexagonal = triangular lattice</a>
%F A003051 a(n) = Sum_{ m^2 | n } A003050(n/m^2).
%F A003051 a(n) = (A000203 + 2*A002324 + 3*A145390)/6. [Rutherford] - N. J. A. Sloane, 
               Mar 13 2009
%Y A003051 Cf. A003050, A054384, A001615, A006984, A054345.
%Y A003051 Sequence in context: A030582 A036762 A032154 this_sequence A097352 A076050 
               A130799
%Y A003051 Adjacent sequences: A003048 A003049 A003050 this_sequence A003052 A003053 
               A003054
%K A003051 nonn,nice,easy
%O A003051 1,3
%A A003051 N. J. A. Sloane (njas(AT)research.att.com).