A232561 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 3*x are in S, and duplicates are deleted as they occur.

1, 2, 3, 6, 4, 9, 7, 18, 5, 12, 10, 27, 8, 21, 19, 54, 15, 13, 36, 11, 30, 28, 81, 24, 22, 63, 20, 57, 55, 162, 16, 45, 14, 39, 37, 108, 33, 31, 90, 29, 84, 82, 243, 25, 72, 23, 66, 64, 189, 60, 58, 171, 56, 165, 163, 486, 17, 48, 46, 135, 42, 40, 117, 38

Offset: 1

Author

Clark Kimberling, Nov 26 2013

Keywords

• nonn
• easy

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 3*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) = (2,3), g(3) = (6,4,9), g(4) = (7,18,5,8,10,27), etc. Concatenating these gives A232561, a permutation of the positive integers. The number of numbers in g(n) is A001590(n), the n-th tribonacci 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 3*x if 3*x has not already occurred.

Examples

			Each x begets x + 1 and 3*x, but if either has already occurred it is deleted.  Thus, 1 begets 2 and 3; in the next generation, 2 begets only 6, and 3 begets 4 and 9.
		

Links

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

Crossrefs

Cf. A232560, A232562, A001590.

Programs

Mathematica

z = 8; g[1] = {1}; g[2] = {2, 3};
g[n_] := Riffle[g[n - 1] + 1, 3 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}]]  (* A232561 *)
Table[Length[g1[n]], {n, 1, z}]  (* A001590 *)
t1 = Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232562 *)