A228897 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.

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, 48, 52, 54, 55, 60, 63, 66, 68, 70, 72, 75, 84, 88, 89, 90, 96, 102, 104, 108, 110, 112, 117, 126, 130, 135, 136, 138, 144, 145, 150, 160, 165, 168, 174, 176, 178

Offset: 1

Author

Clark Kimberling, Sep 08 2013

Keywords

• nonn
• easy

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.

Examples

			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,...).
		

Crossrefs

Cf. A228853.

Programs

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]]