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Tree read by levels generated by these rules: 1 is at the top, and the node x has child nodes x+1, 2*x, and 3*x, where duplicates are deleted as they occur.
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%I #14 Nov 26 2016 11:37:43

%S 1,2,3,4,6,9,5,8,12,7,18,10,27,15,16,24,13,36,14,21,19,54,11,20,30,28,

%T 81,45,17,32,48,25,72,26,39,37,108,42,22,63,38,57,55,162,33,40,60,31,

%U 90,29,56,84,82,243,46,135,34,51,64,96,49,144,50,75,73

%N Tree read by levels generated by these rules: 1 is at the top, and the node x has child nodes x+1, 2*x, and 3*x, where duplicates are deleted as they occur.

%C 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), g(2) = (2,3), g(3) = (4,6,9), g(4) = (5,8,12,7,18,10,27), ... Concatenating gives 1,2,3,4,6,9,5,...

%H Clark Kimberling, <a href="/A233559/b233559.txt">Table of n, a(n) for n = 1..5000</a>

%e To generate S, start with g(1) = (1). Then 1 begets 2 and 3; then 2 begets 4 and 3 begets 6 and 9.

%t x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 2 x, 3 x}]]], {8}]; x (* A233559 *)

%t y = Flatten[Table[Position[x, n], {n, 1, 157}]] (* A233560 *)

%Y Cf. A232559, A233560.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Dec 14 2013

%E Name edited by _Ivan Neretin_, Nov 26 2016