OFFSET
1,2
COMMENTS
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.
EXAMPLE
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,...).
MATHEMATICA
f[x_, y_] := {{y, x + y}, {y, x* y}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 08 2013
STATUS
approved