A224361 - OEIS

%I #13 Sep 14 2017 09:38:43

%S 1,2,2,5,3,5,3,1,6,10,5,4,8,11,5,6,4,7,12,6,9,5,2,1,12,2,10,15,4,7,9,

%T 14,7,9,16,5,12,5,10,13,4,11,6,6,18,5,5,8,1,3,12,17,5,19,37,7,5,15,13,

%U 5,10,17,7,10,38,14,7,6,2,17,8,14,7,2,20,17,15

%N The length of the Collatz (3k+1) sequence for all odd negative fractions and integers.

%C This sequence is the extension of A210688 with negative values.

%C We consider the triangle T(n,k) = -(n-k)/(2k+1) for n = 1,2,... and k = 0..n-1.

%C The example shown below gives a general idea of this regular triangle. This contains all negative 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 The initial triangle T(n,k) begins

%C   -1;

%C   -2, -1/3;

%C   -3, -2/3, -1/5;

%C   -4, -3/3, -2/5, -1/7;

%C   -5, -4/3, -3/5, -2/7, -1/9;

%C   -6, -5/3, -4/5, -3/7, -2/9, -1/11;

%C   ...

%C Needs a more precise definition. - _N. J. A. Sloane_, Sep 14 2017

%F a(n) = A224360(n) + 1.

%e The triangle of lengths begins

%e   1;

%e   2, 2;

%e   5, 3, 5;

%e   3, 1, 6, 10;

%e   5, 4, 8, 11, 5;

%e   ...

%e Individual numbers have the following Collatz sequences (including the first term):

%e [-1] => [1] because -1 -> -1 with 1 iteration;

%e [-2 -1/3] => [2, 2] because: -2 -> -1 => 2 iterations; -1/3 -> 0 => 2 iterations;

%e [-3 -2/3 -1/5] => [5, 3, 5] because: -3 -> -8 -> -4 -> -2 -> -1 => 5 iterations; -2/3 -> -1/3 -> 0 => 3 iterations; -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 => 5 iterations.

%t Collatz2[n_] := Module[{lst = NestWhileList[If[EvenQ[Numerator[#]], #/2, 3 # + 1] &, n, Unequal, All]}, If[lst[[-1]] == -1, lst = Drop[lst, -2], 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] , {n, 13}, {k, 0, n - 1}]; Flatten[t] (* program from _T. D. Noe_, adapted for this sequence - see  A210688 *).

%Y Cf. A210516, A210688, A224299, A224300, A224360.

%K nonn

%O 1,2

%A _Michel Lagneau_, Apr 04 2013