

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
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OFFSET

1,2


COMMENTS

This sequence is the extension of A210688 with negative values.
We consider the triangle T(n,k) = (nk)/(2k+1) for n = 1,2,... and k = 0..n1.
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



