A232895 Sequence (or tree) S of all positive integers in the order generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur.

1, 2, 3, 4, 5, 6, 8, 7, 10, 12, 16, 9, 14, 20, 24, 18, 32, 11, 28, 22, 40, 26, 48, 36, 34, 64, 13, 30, 56, 44, 42, 80, 52, 50, 96, 38, 72, 68, 66, 128, 15, 60, 58, 112, 46, 88, 84, 82, 160, 54, 104, 100, 98, 192, 76, 74, 144, 70, 136, 132, 130, 256, 17, 62

Offset

1,2

Comments

Let S be the sequence (or tree) of numbers generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur. Every positive integer occurs exactly once in S, so that S is a permutation of the natural numbers. Deleting duplicates as they occur, the generations of S are given by g(1) = (1,2), g(2) = (3,4), g(3) = (5,6,8), g(4) = (7,10,12,16), ... Concatenating gives 1,2,3,4,5,6,8,... Conjecture: the position of the n-th odd positive integer in S is the linearly recurrent sequence given by A232896(n) for n>=1.

Links

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

Example

To generate S, start with g(1) = (1,2).  Then 1 begets 3 and 2, but 2 is deleted as a duplicate, and 2 begets 4 and 4, of which the second 4 is deleted; thus g(2) = (3,4).

Mathematica

x = {1, 2}; dx = 0; Do[x = DeleteDuplicates[Flatten[AppendTo[x, Transpose[{# + 2, 2*#}] &[Drop[x, Length[x] - dx]]]]]; dx = Length[x] - dx, {31}]; x  (* A232895 *)
t = Flatten[Position[Denominator[x/2], 2]] (* A232896 conjectured *)
(* Peter J. C. Moses, Dec 02 2013 *)

Crossrefs

Cf. A232559, A232896.

Sequence in context: A366948 A297440 A362134 * A274607 A339607 A262374

Adjacent sequences: A232892 A232893 A232894 * A232896 A232897 A232898

Keyword

nonn,easy

Author

Clark Kimberling, Dec 02 2013

Status

approved