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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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%I #6 Jan 05 2019 18:17:26

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%e n = 5 -> 33,34,17,18,9,10,5,6,3,4,2,1

%p 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" "); ) ) }

%t Table[NestWhileList[If[EvenQ[#],#/2,#+1]&,2^n+1,#!=1&],{n,10}]//Flatten (* _Harvey P. Dale_, Jan 05 2019 *)

%K easy,nonn,tabf

%O 1,1

%A _Cino Hilliard_, Apr 19 2003

%E Corrected and extended by _Harvey P. Dale_, Jan 05 2019