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%I #7 Dec 04 2016 19:46:33
%S 1,2,5,4,10,25,8,20,50,125,16,40,100,250,625,32,80,200,500,1250,3125,
%T 64,160,400,1000,2500,6250,15625,128,320,800,2000,5000,12500,31250,
%U 78125,256,640,1600,4000,10000,25000,62500,156250,390625,512,1280,3200,8000
%N Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2*x and 5*x + 3 are in S, and duplicates are deleted as they occur.
%C Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then 2*x and 5*x are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,5), g(3) = (4,10,25), etc. Concatenating these gives A232646, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is n. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to 2*x if 2*x has not already occurred, and an edge from x to 3*x if 3*x has not already occurred.
%H Clark Kimberling, <a href="/A232646/b232646.txt">Table of n, a(n) for n = 1..1000</a>
%F Counting the top row as row 0 and writing <i,j> for (2^i)*(5*j) , the numbers in row n are <n,0>, <n-1,1>, ..., <0,n>.
%e Each x begets 2*x and 5*x, but if either has already occurred it is deleted. Thus, 1 begets 2 and 5; then 2 begets 4 and 10, and 5 begets only 25, so that g(3) = (4,10,25). Writing generations as rows results in a triangle whose first five rows are as follows:
%e 1
%e 2 .... 5
%e 4 .... 10 ... 25
%e 8 .... 20 ... 50 ... 125
%e 16 ... 40 ... 100 .. 250 .. 625
%t x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x (* _Peter J. C. Moses_, Nov 27 2013 *)
%Y Cf. A232559, A232639, A036561.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Nov 28 2013
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