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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. 36
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 (list; graph; refs; listen; history; text; internal format)
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 n-th 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: A265734 A232560 A183090 * A094138 A116538 A084287

Adjacent sequences:  A232556 A232557 A232558 * A232560 A232561 A232562

KEYWORD

nonn,easy,changed

AUTHOR

Clark Kimberling, Nov 26 2013

STATUS

approved

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Last modified August 17 06:03 EDT 2017. Contains 290635 sequences.