Looks repetitive but isn't

Suppose you start with 13. Then if the number you have is smaller than 16 double it, if it is larger than 16 divide it by 3. If it gets to exactly 16 stop. Otherwise repeat with this new number you reach. Here are the first few:

13
26.0
8.66666666667
17.3333333333
5.77777777778
11.5555555556
23.1111111111
7.7037037037
15.4074074074
30.8148148148
10.2716049383
20.5432098765
6.84773662551
13.695473251
27.3909465021
9.13031550069
18.2606310014
6.08687700046
12.1737540009


Here are the first 800 numbers of the series plotted in this 800 pixel wide image. The height of each is rounded to the nearest pixel.
(You really have to click and zoom in to be able to see the pattern)

Isn't that interesting there is a clear regularity to the pattern but I have verified that the heights of any two of these pixels are no closer than .005 apart. That means in the first 800 of the series you never repeat exactly a value. So it can't be a simple repetitive pattern.