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A232644 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 3 are in S, and duplicates are deleted as they occur.

%I A232644 #7 Aug 20 2017 23:20:39
%S A232644 1,2,5,3,7,6,13,4,9,8,17,15,14,29,11,10,21,19,18,37,16,33,31,30,61,12,
%T A232644 25,23,22,45,20,41,39,38,77,35,34,69,32,65,63,62,125,27,26,53,24,49,
%U A232644 47,46,93,43,42,85,40,81,79,78,157,36,73,71,70,141,67,66
%N A232644 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 3 are in S, and duplicates are deleted as they occur.
%C A232644 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 + 3 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) = (2,5), g(3) = (3,7,6,13), etc. Concatenating these gives A232644, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is L(n), where F = A000032, the Lucas 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 + 3 if 2*x + 3 has not already occurred.
%H A232644 Clark Kimberling, <a href="/A232644/b232644.txt">Table of n, a(n) for n = 1..1000</a>
%e A232644 Each x begets x + 1 and 2*x + 3, but if either has already occurred it is deleted. Thus, 1 begets 2 and 5; then 2 begets 3 and 7, and 5 begets 6 and 13, so that g(3) = (3,7,6,13).
%t A232644 z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 3]; 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}]]  (* A232644 *)
%t A232644 Table[Length[g1[n]], {n, 1, z}]  (* A000032 *)
%t A232644 Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232645 *)
%Y A232644 Cf. A000032, A232559, A232639, A232645.
%K A232644 nonn,easy
%O A232644 1,2
%A A232644 _Clark Kimberling_, Nov 28 2013