cp's OEIS Frontend

A378376 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.

%I A378376 #15 Jan 01 2025 00:15:11
%S A378376 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 A378376 41,81,161,55,109,37,73,25,49,17,33,65,129,257,87,173,341,117,225,455,
%U A378376 153,305,607,209,405,809,273,543,185,369,721,1433,481,961,321
%N A378376 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 A378376 The number of steps required is A261870(x) so that a(n) = x is the smallest x where A261870(x) = n.
%C A378376 a(n) <= 2^n is a simple upper bound, since x = 2^n requires n steps to reach 1.
%C A378376 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 A378376 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 A378376 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 A378376 Kevin Ryde, <a href="/A378376/b378376.txt">Table of n, a(n) for n = 0..130</a>
%H A378376 Kevin Ryde, <a href="/A261870/a261870_1.c.txt">C Code</a>
%e A378376 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 A378376 a(4) = 3 is also an example where a(n) is its lower bound (a(n-1)+1)/3 (with a(3) = 8).
%o A378376 (C) /* See links. */
%Y A378376 Cf. A261870.
%K A378376 nonn
%O A378376 0,2
%A A378376 _Kevin Ryde_, Nov 25 2024