A228898 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^2 + y^2) are edges.

1, 2, 3, 5, 7, 8, 12, 13, 16, 19, 21, 29, 31, 34, 39, 45, 50, 55, 63, 73, 74, 81, 89, 97, 112, 119, 131, 144, 155, 160, 178, 185, 186, 191, 193, 205, 212, 233, 236, 246, 257, 283, 297, 312, 343, 369, 377, 391, 398, 417, 425, 441, 469, 479, 482, 505, 524, 555

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)->(2,5)->(5,29)->(29,866)->... These branches contribute to A228898, as subsequences, the Fibonacci numbers, A000045, and A000283.

Examples

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,13), (5,7), (5,29)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 5, 7, 8, 12, 13, 16, 19,...).
		

Crossrefs

Cf. A228853, A000045.

Programs

Mathematica

f[x_, y_] := {{y, x + y}, {y, x^2 + y^2}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {18}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]