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

Author

Clark Kimberling, Dec 02 2013

Keywords

• nonn
• easy

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.

Examples

			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).
		

Links

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

Crossrefs

Cf. A232559, A232896.

Programs

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 *)