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

1,2

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

Links

Table of n, a(n) for n=1..58.

Example

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,...).

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]]

Crossrefs

Cf. A228853, A000045.

Sequence in context: A090704 A112055 A291485 * A352695 A372050 A076386

Adjacent sequences: A228895 A228896 A228897 * A228899 A228900 A228901

Keyword

nonn,easy

Author

Clark Kimberling, Sep 08 2013

Status

approved