

A228853


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.


9



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, 49, 50, 55, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 79, 80, 81, 89, 97, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 115, 116, 117, 119, 121, 123, 128
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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.
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.)
Is A228853 essentially A141832? (If so, the answer to the question in Comments at A141832 is that A141832 is infinite.)


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000


EXAMPLE

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, ...}.
Part 2: The branch 2, 3, 8, 11, 19, 30, 49, 128, 305 has edgelabels 1, 2, 1, 1, 1, 1, 2, 2, so that 305/128 = [2, 2, 1, 1, 1, 1, 2, 1, 2].


MATHEMATICA

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]]


CROSSREFS

Cf. A141832, A228854, A228855, A228856, A000045, A000129.
Sequence in context: A026424 A298207 A229125 * A141832 A066680 A298865
Adjacent sequences: A228850 A228851 A228852 * A228854 A228855 A228856


KEYWORD

nonn,easy,nice


AUTHOR

Clark Kimberling, Sep 05 2013


STATUS

approved



