

A232640


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, 5, 4, 7, 6, 11, 9, 8, 15, 13, 12, 23, 10, 19, 17, 16, 31, 14, 27, 25, 24, 47, 21, 20, 39, 18, 35, 33, 32, 63, 29, 28, 55, 26, 51, 49, 48, 95, 22, 43, 41, 40, 79, 37, 36, 71, 34, 67, 65, 64, 127, 30, 59, 57, 56, 111, 53, 52, 103, 50, 99, 97, 96, 191
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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) = (2,3), g(3) = (5,4,7), etc. Concatenating these gives A232640, a permutation of the positive integers. The number of numbers in g(n) is F(n), 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 and 3; then 2 begets only 5, and 3 begets (4,7), so that g(3) = (5,4,7).


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}]] (* A232640 *)
Table[Length[g1[n]], {n, 1, z}] (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232641 *)


CROSSREFS

Cf. A000045, A232559, A232639, A232641.
Sequence in context: A293977 A282649 A102454 * A085790 A257339 A210882
Adjacent sequences: A232637 A232638 A232639 * A232641 A232642 A232643


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 28 2013


STATUS

approved



