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%I #9 Mar 12 2014 00:53:28
%S 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,
%T 13,30,56,44,42,80,52,50,96,38,72,68,66,128,15,60,58,112,46,88,84,82,
%U 160,54,104,100,98,192,76,74,144,70,136,132,130,256,17,62
%N 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.
%C 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.
%H Clark Kimberling, <a href="/A232895/b232895.txt">Table of n, a(n) for n = 1..1000</a>
%e 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).
%t 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 t = Flatten[Position[Denominator[x/2], 2]] (* A232896 conjectured *)
%t (* _Peter J. C. Moses_, Dec 02 2013 *)
%Y Cf. A232559, A232896.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Dec 02 2013
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