

A232563


Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 4*x are in S, and duplicates are deleted as they occur.


4



1, 2, 4, 3, 8, 5, 16, 12, 9, 32, 6, 20, 17, 64, 13, 48, 10, 36, 33, 128, 7, 24, 21, 80, 18, 68, 65, 256, 14, 52, 49, 192, 11, 40, 37, 144, 34, 132, 129, 512, 28, 25, 96, 22, 84, 81, 320, 19, 72, 69, 272, 66, 260, 257, 1024, 15, 56, 53, 208, 50, 196, 193, 768
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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 4*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) = (2,4), g(3) = (3,8,5,16), g(4) = (12,9,32,6,20,17,64), etc. Concatenating these gives A232563, a permutation of the positive integers. The number of numbers in g(n) is A001631(n), the nth tetranacci 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 4*x if 4*x has not already occurred.


LINKS

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


EXAMPLE

Each x begets x + 1 and 4*x, but if either has already occurred it is deleted. Thus, 1 begets 2 and 4; in the next generation, 2 begets 3 and 8, and 4 begets 5 and 16.


MATHEMATICA

z = 8; g[1] = {1}; g[2] = {2, 4}; g[n_] := Riffle[g[n  1] + 1, 4 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}]] (* A232563 *)
Table[Length[g1[n]], {n, 1, z}] (* A001631 *)
t1 = Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232564 *)


CROSSREFS

Cf. A232559, A232564, A001631.
Sequence in context: A120242 A322864 A054427 * A048672 A277517 A248513
Adjacent sequences: A232560 A232561 A232562 * A232564 A232565 A232566


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 26 2013


STATUS

approved



