A224361 The length of the Collatz (3k+1) sequence for all odd negative fractions and integers.

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

Offset: 1

Author

Michel Lagneau, Apr 04 2013

Keywords

• nonn
• tabl

Comments

This sequence is the extension of A210688 with negative values.

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

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).

The initial triangle T(n,k) begins

-1;

-2, -1/3;

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

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

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

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

...

Needs a more precise definition. - N. J. A. Sloane, Sep 14 2017

Examples

			The triangle of lengths begins
  1;
  2, 2;
  5, 3, 5;
  3, 1, 6, 10;
  5, 4, 8, 11, 5;
  ...
Individual numbers have the following Collatz sequences (including the first term):
  [-1] => [1] because -1 -> -1 with 1 iteration;
  [-2 -1/3] => [2, 2] because: -2 -> -1 => 2 iterations; -1/3 -> 0 => 2 iterations;
  [-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.
		

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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