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%I #9 Dec 02 2024 21:18:50
%S 1,2,4,8,3,6,11,22,43,15,29,10,20,7,14,5,9,18,35,13,23,46,91,31,61,21,
%T 41,81,161,55,109,37,73,25,49,17,33,65,129,257,87,173,341,117,225,455,
%U 153,305,607,209,405,809,273,543,185,369,721,1433,481,961,321
%N Smallest starting x which requires n steps to reach 1 under the map x -> 3x-1 if x odd, x -> 3x-1 or x/2 if x even.
%C The number of steps required is A261870(x) so that a(n) = x is the smallest x where A261870(x) = n.
%C a(n) <= 2^n is a simple upper bound, since x = 2^n requires n steps to reach 1.
%C But 2*a(n-1) = x is not an upper bound on a(n), since although x/2 = a(n-1) requires a further n-1 steps, x can also step to 3x-1 and doing so might be fewer steps (which it is for example at n=45).
%C a(n) >= (a(n-1)+1)/3 is a lower bound since a(n) = x must have 3x-1 >= a(n-1) so as to reach somewhere requiring n-1 further steps.
%C If a(n-1) == 2 (mod 6), then equality a(n) = (a(n-1)+1)/3 holds since then a(n) is odd and its first step must be 3x-1 (as for example at n=4).
%H Kevin Ryde, <a href="/A378376/b378376.txt">Table of n, a(n) for n = 0..120</a>
%H Kevin Ryde, <a href="/A261870/a261870.c.txt">C Code</a>
%e For n=4, a(4) = 3 is the smallest x requiring n=4 steps to reach 1 (by trajectory 3 -> 8 -> 4 -> 2 -> 1).
%e a(4) = 3 is also an example where a(n) is its lower bound (a(n-1)+1)/3 (with a(3) = 8).
%o (C) /* See links. */
%Y Cf. A261870.
%K nonn,new
%O 0,2
%A _Kevin Ryde_, Nov 25 2024