

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.


2



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, 25, 23, 22, 45, 20, 41, 39, 38, 77, 35, 34, 69, 32, 65, 63, 62, 125, 27, 26, 53, 24, 49, 47, 46, 93, 43, 42, 85, 40, 81, 79, 78, 157, 36, 73, 71, 70, 141, 67, 66
(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 + 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.


LINKS



EXAMPLE

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).


MATHEMATICA

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 *)
Table[Length[g1[n]], {n, 1, z}] (* A000032 *)
Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232645 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



