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A232646
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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.
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1
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1, 2, 5, 4, 10, 25, 8, 20, 50, 125, 16, 40, 100, 250, 625, 32, 80, 200, 500, 1250, 3125, 64, 160, 400, 1000, 2500, 6250, 15625, 128, 320, 800, 2000, 5000, 12500, 31250, 78125, 256, 640, 1600, 4000, 10000, 25000, 62500, 156250, 390625, 512, 1280, 3200, 8000
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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>.
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EXAMPLE
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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:
1
2 .... 5
4 .... 10 ... 25
8 .... 20 ... 50 ... 125
16 ... 40 ... 100 .. 250 .. 625
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MATHEMATICA
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x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x (* Peter J. C. Moses, Nov 27 2013 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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