

A232559


Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x are in S, and duplicates are deleted as they occur.


39



1, 2, 3, 4, 6, 5, 8, 7, 12, 10, 9, 16, 14, 13, 24, 11, 20, 18, 17, 32, 15, 28, 26, 25, 48, 22, 21, 40, 19, 36, 34, 33, 64, 30, 29, 56, 27, 52, 50, 49, 96, 23, 44, 42, 41, 80, 38, 37, 72, 35, 68, 66, 65, 128, 31, 60, 58, 57, 112, 54, 53, 104, 51, 100, 98, 97
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OFFSET

1,2


COMMENTS

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x are in S. Then S is the set of all positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2), g(3) = (3,4), g(4) = (6,5,8), g(5) = (7,12,10,9,16), etc. Concatenating these gives A232559, a permutation of the positive integers. The number of numbers in g(n) is A000045(n), the nth Fibonacci number. It is helpful to show the results as a tree with the terms of S as nodes and edges from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x if 2*x has not already occurred. The positions of the odd numbers are given by A026352, and of the evens, by A026351.
The previously mentioned tree is an example of a fractal tree; that is, an infinite rooted tree T such that every complete subtree of T contains a subtree isomorphic to T.  Clark Kimberling, Jun 11 2016
The similar sequence S', generated by these rules: 0 is in S', and if x is in S', then 2*x and x+1 are in S', and duplicates are deleted as they occur, appears to equal A048679.  Rémy Sigrist, Aug 05 2017


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Clark Kimberling)


EXAMPLE

Each x begets x + 1 and 2*x, but if either has already occurred it is deleted. Thus, 1 begets 2, which begets (3,4); from which 3 begets only 6, and 4 begets (5,8).


MAPLE

a:= proc() local l, s; l, s:= [1], {1}:
proc(n) option remember; local i, r; r:= l[1];
l:= subsop(1=NULL, l);
for i in [1+r, r+r] do if not i in s then
l, s:=[l[], i], s union {i} fi
od; r
end
end():
seq(a(n), n=1..100); # Alois P. Heinz, Aug 06 2017


MATHEMATICA

z = 12; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n  1] + 1, 2 g[n  1]]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n  1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n  1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]] (* A232559 *)
Table[Length[g1[n]], {n, 1, z}] (* Fibonacci numbers *)
t1 = Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232560 *)


CROSSREFS

Cf. A232560, A232561, A232563, A226080, A226130, A000045, A026352, A026351, A048679.
Sequence in context: A299759 A232560 A183090 * A094138 A116538 A084287
Adjacent sequences: A232556 A232557 A232558 * A232560 A232561 A232562


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 26 2013


STATUS

approved



