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A081246
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Triangle in which (2^n+1)st row gives trajectory of x=2^n+1 under the map x -> x/2 if x is even, x -> x+1 if x is odd, stopping when reaching 1.
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0
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3, 4, 2, 1, 5, 6, 3, 4, 2, 1, 9, 10, 5, 6, 3, 4, 2, 1, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1, 129, 130, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1, 257, 258, 129, 130, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1, 513, 514, 257, 258, 129, 130, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1
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OFFSET
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1,1
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COMMENTS
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This is the 2^n+1 conjecture and is easily proved to converge to 1. The number of steps required to reach 1 is always 2n+2. Since (2^(n)+1+1)/2 = 2^(n-1)+1 (2^(n-1)+1+1)/2 = 2^(n-2)+1 .... (2^(n-n+1)+1+1)/2 = 2^(n-n)+1 = 2 2/2 = 1 thus 1 is guaranteed.
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LINKS
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EXAMPLE
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n = 5 -> 33,34,17,18,9,10,5,6,3,4,2,1
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MAPLE
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pxpr(n) = { for(x=1, n, x1=2^x+1; print1(x1" "); while(x1>1, if(x1%2==0, x1/=2, x1 = x1+1); print1(x1" "); ) ) }
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MATHEMATICA
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Table[NestWhileList[If[EvenQ[#], #/2, #+1]&, 2^n+1, #!=1&], {n, 10}]//Flatten (* Harvey P. Dale, Jan 05 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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