%I A003050 M0229
%S A003050 1,1,2,2,2,3,3,4,3,4,3,6,4,5,6,6,4,7,5,8,8,7,5,12,6,
%T A003050 8,7,10,6,14,7,10,10,10,10,14,8,11,12,16,8,18,9,14,14,13,9,20,11,
%U A003050 16,14,16,10,19,14,20,16,16,11,28,12,17,18,18,16,26,13,20,18,26,13,28
%N A003050 Number of primitive sublattices of index n in hexagonal lattice: triples 
               x,y,z from Z/nZ with x+y+z=0, discarding any triple that can be obtained 
               from another by multiplying by a unit and permuting.
%C A003050 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%C A003050 Also the number of triangles with vertices on points of the hexagonal 
               lattice that have area equal to n/2. - Amihay Hanany, Oct 15 2009
%D A003050 A. Altshuler, Construction and enumeration of regular maps on the torus, 
               Discrete Math. 4 (1973), 201-217.
%D A003050 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003050 T. D. Noe, <a href="b003050.txt">Table of n, a(n) for n=1..1000</a>
%H A003050 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 A003050 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 A003050 <a href="Sindx_Aa.html#A2">Index entries for sequences related to A2 
               = hexagonal = triangular lattice</a>
%H A003050 <a href="Sindx_Su.html#sublatts">Index entries for sequences related 
               to sublattices</a>
%F A003050 Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6) * n prod_{i=1..w} 
               (1 + 1/p_i) + (C_1)/3 + 2^(w-2+C_2), where
%F A003050 C_2 = 2 if n == 0 mod 8, 1 if n == 1, 3, 4, 5, 7 mod 8, 0 if n == 2, 
               6 mod 8.
%F A003050 C_1 = 0 if 2|n or 9|n, = prod_{i=1..w, p_i > 3} ( 1+ Legendre(p_i, 3)) 
               otherwise and
%e A003050 For n = 6 the 3 primitive triples are 510, 411, 321.
%t A003050 Join[{1}, Table[p=Transpose[FactorInteger[n]][[1]]; If[Mod[n,2]==0 || 
               Mod[n,9]==0, c1=0, c1=Product[(1+JacobiSymbol[p[[i]],3]), {i,Length[p]}]]; 
               c2={2,1,0,1,1,1,0,1}[[1+Mod[n,8]]]; n*Product[(1+1/p[[i]]), {i, Length[p]}]/
               6+c1/3+2^(Length[p]-2+c2), {n,2,100}]] [From T. D. Noe (noe(AT)sspectra.com), 
               Oct 18 2009]
%Y A003050 Cf. A003051, A001615, A006984, A007997, A048259, A054345.
%Y A003050 Sequence in context: A038809 A078342 A107325 this_sequence A070868 A155216 
               A064144
%Y A003050 Adjacent sequences: A003047 A003048 A003049 this_sequence A003051 A003052 
               A003053
%K A003050 nonn,nice
%O A003050 1,3
%A A003050 N. J. A. Sloane (njas(AT)research.att.com).