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

1,2

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 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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A232560, A232562, A001590.

Sequence in context: A103864 A268712 A152679 * A348395 A320123 A363444

Adjacent sequences: A232558 A232559 A232560 * A232562 A232563 A232564

Keyword

nonn,easy

Author

Clark Kimberling, Nov 26 2013

Status

approved