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A224361 The length of the Collatz (3k+1) sequence for all odd negative fractions and integers. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A286109 A239665 A178179

Adjacent sequences:  A224358 A224359 A224360 * A224362 A224363 A224364

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 04 2013

STATUS

approved

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Last modified June 19 10:32 EDT 2019. Contains 324219 sequences. (Running on oeis4.)