%I #30 Oct 22 2018 04:57:50
%S 1,2,3,5,7,8,11,12,13,17,18,19,21,26,27,29,30,31,34,41,43,44,45,46,47,
%T 49,50,55,63,64,65,67,68,69,70,71,73,74,75,76,79,80,81,89,97,99,100,
%U 101,104,105,106,108,109,111,112,115,116,117,119,121,123,128
%N Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.
%C As a tree, infinitely many branches are essentially linearly recurrent sequences. The extreme cases, (1,2) -> (2,3) -> (3,5) -> ... and (1,2) -> (2,5) -> (5,12) -> ..., contribute A000045 (Fibonacci numbers) and A000129 (Pell numbers) to A228853.
%C Suppose that (u,v) and (v,w) are consecutive edges. The continued fraction of w/v is obtained from the continued fraction of v/u by prefixing 1 if w = v + u, or 2 if w = 2v + u. Consequently, if each edge is labeled with 1 or 2 in the obvious way, then the continued fraction of w/v is the sequence of 1s and 2s, in reverse order, from the node 2 to the node w, with 2 attached at the end. (See Example, Part 2.)
%C Is A228853 essentially A141832? (If so, the answer to the question in Comments at A141832 is that A141832 is infinite.)
%C Yes; the initial node (1,2) adds a single 2 to the end of the fraction, and subsequent edges prepend 1's and 2's. - _Charlie Neder_, Oct 21 2018
%H Ivan Neretin, <a href="/A228853/b228853.txt">Table of n, a(n) for n = 1..10000</a>
%e Part 1: Taking the first generation of edges of the tree to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,8), (5,7), (5,12)}, ... Expelling duplicate nodes and sorting leave {1, 2, 3, 5, 7, 8, ...}.
%e Part 2: The branch 2, 3, 8, 11, 19, 30, 49, 128, 305 has edge-labels 1, 2, 1, 1, 1, 1, 2, 2, so that 305/128 = [2, 2, 1, 1, 1, 1, 2, 1, 2].
%t f[x_, y_] := {{y, x + y}, {y, x + 2 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]]
%Y Cf. A141832, A228854, A228855, A228856, A000045, A000129.
%K nonn,easy,nice
%O 1,2
%A _Clark Kimberling_, Sep 05 2013
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