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

1,2

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

Links

Table of n, a(n) for n=1..77.

Formula

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

Example

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.

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 *).

Crossrefs

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

Sequence in context: A157223 A174608 A130327 * A341443 A224166 A286109

Adjacent sequences: A224358 A224359 A224360 * A224362 A224363 A224364

Keyword

nonn

Author

Michel Lagneau, Apr 04 2013

Status

approved