%I
%S 1,2,3,4,5,6,8,9,10,12,13,15,16,18,20,21,24,26,30,32,34,35,39,40,42,
%T 48,52,54,55,60,63,66,68,70,72,75,84,88,89,90,96,102,104,108,110,112,
%U 117,126,130,135,136,138,144,145,150,160,165,168,174,176,178
%N Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x+y) and (y,x*y) are edges.
%C The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). The extreme branches are (1,2)>(2,3)>(3,5)>(5,8)>... and (1,2)>(2,4)>(4,8)>(8,32)>... These branches contribute to A228897, as subsequences, the Fibonacci numbers, A000045, and the sequence 2^(A000045) = A000302.
%e Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,4)}, which grows G(3) = {(3,5), (3,6), (4,6), (4,8)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 4, 5, 6, 8, 9, 10, 12,...).
%t f[x_, y_] := {{y, x + y}, {y, x* y}}; x = 1; y = 2; t = {{x, y}};
%t u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
%t w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
%t Sort[Union[w]]
%Y Cf. A228853.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Sep 08 2013
