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A210516 The length-1 of the Collatz (3k+1) sequence for all odd fractions and integers.

%I A210516 #33 Jun 13 2017 03:53:42
%S A210516 0,1,2,7,3,3,2,0,3,6,5,4,15,7,5,8,9,3,11,6,7,16,1,0,8,2,7,4,3,4,16,5,
%T A210516 7,25,4,17,19,5,13,12,6,7,17,18,8,6,7,3,0,3,22,4,3,8,31,14,10,6,9,11,
%U A210516 26,12,19,21,32,10,9,10,1,31,8,7,18,2,8,16,11,76
%N A210516 The length-1 of the Collatz (3k+1) sequence for all odd fractions and integers.
%C A210516 This sequence is the unification, in the limit, of the length of Collatz sequences for all fractions whose denominator is odd, and also all integers.
%C A210516 The sequence A210483 gives the triangle read by rows giving trajectory of k/(2n+1) in Collatz problem, k = 1..2n, but particular attention should be paid to numbers in the triangle T(n,k) = (n-k)/(2k+1) for n = 1,2,... and k = 0..n-1.
%C A210516 The example shown below gives a general idea of this regular triangle. This contains all fractions whose denominator is odd and all integers. Now, from T(n,k) we could introduce a 3D triangle in order to produce a complete Collatz sequence starting from each rational T(n,k).
%C A210516 Remark: a(A000124(n)) = A006577(n) because the first column of this triangle generates A006577.
%C A210516 The triangle T(n,k) begins
%C A210516   1;
%C A210516   2,  1/3;
%C A210516   3,  2/3,  1/5;
%C A210516   4,  3/3,  2/5,  1/7;
%C A210516   5,  4/3,  3/5,  2/7,  1/9;
%C A210516   6,  5/3,  4/5,  3/7,  2/9, 1/11;
%C A210516   ...
%H A210516 Michel Lagneau, <a href="/A210516/b210516.txt">Rows n = 1..100, flattened</a>
%H A210516 J. C. Lagarias, <a href="http://pldml.icm.edu.pl:80/mathbwn/element/bwmeta1.element.bwnjournal-article-aav56i1p33bwm?q=bwmeta1.element.bwnjournal-number-aa-1990-56-1&amp;qt=CHILDREN-STATELESS">The set of rational cycles for the 3x+1 problem,</a> Acta Arith. 56 (1990), 33-53.
%e A210516 The triangle of lengths begins
%e A210516   0;
%e A210516   1,  2;
%e A210516   7,  3,  3;
%e A210516   2,  0,  3,  6;
%e A210516   5,  4, 15,  7,  5;
%e A210516   ...
%e A210516 Individual numbers have the following Collatz sequences:
%e A210516 [1] => [0] (0 iteration);
%e A210516 [2  1/3] => [1, 2] because: 2 -> 1  => 1 iteration;  1/3 -> 2 -> 1 => 2 iterations;
%e A210516 [3  2/3  1/5] => [7, 3, 3] because: 3->10->5->16->8->4->2->1 => 7 iterations; 2/3 -> 1/3 -> 2 -> 1 => 3 iterations; 1/5 -> 8/5 -> 4/5 -> 2/5 => 3 iterations.
%t A210516 Collatz2[n_] := Module[{lst = NestWhileList[If[EvenQ[Numerator[#]], #/2, 3 # + 1] &, n, Unequal, All]}, If[lst[[-1]] == 1, lst = Drop[lst, -3], If[lst[[-1]] == 2, lst = Drop[lst, -2], If[lst[[-1]] == 4, lst = Drop[lst, -1], If[MemberQ[Rest[lst], lst[[-1]]], lst = Drop[lst, -1]]]]]]; t = Table[s = Collatz2[(n - k)/(2*k + 1)]; Length[s] - 1, {n, 12}, {k, 0, n - 1}]; Flatten[t] (* _T. D. Noe_, Jan 28 2013 *)
%Y A210516 Cf. A210483, A210468, A210688.
%K A210516 nonn,tabl
%O A210516 1,3
%A A210516 _Michel Lagneau_, Jan 26 2013