

A232638


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


2



1, 2, 3, 4, 5, 7, 6, 9, 8, 13, 11, 10, 17, 15, 14, 25, 12, 21, 19, 18, 33, 16, 29, 27, 26, 49, 23, 22, 41, 20, 37, 35, 34, 65, 31, 30, 57, 28, 53, 51, 50, 97, 24, 45, 43, 42, 81, 39, 38, 73, 36, 69, 67, 66, 129, 32, 61, 59, 58, 113, 55, 54, 105, 52, 101, 99
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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  1 are in S. Then S is the set of 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), g(4) = (4,5), etc. Concatenating these gives A232638, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is F(n1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x  1 if 2*x  1 has not already occurred.


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000


EXAMPLE

Each x begets x + 1 and 2*x  1, but if either has already occurred it is deleted. Thus, 1 begets 2, which begets 3, which begets 4 and 5, which beget 7 and (6,8), respectively.


MATHEMATICA

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n  1] + 1, 2 g[n  1]  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}]] (* A232638 *)
Table[Length[g1[n]], {n, 1, z}] (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232639 *)


CROSSREFS

Cf. A000045, A232559, A232639.
Sequence in context: A183084 A185975 A232639 * A023826 A283192 A266638
Adjacent sequences: A232635 A232636 A232637 * A232639 A232640 A232641


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 28 2013


STATUS

approved



