Repeating cycles in the $3n-1$ problem While tracking sequences beginning with 1-to-3 digit integers, I have found 3 different repeating cycles in the $3n-1$ problem (similar to the Collatz Conjecture). They are 1, 2, 1..., 5, 14, 7, 20, 10, 5..., and 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17....
Has there been much research into this problem? If so, has anyone found any other such sequences? I have checked starting values up to approximately 150.
 A: To spell out in more detail what I hinted at in a comment:
Consider the standard Collatz mapping $x \rightarrow 3x + 1$ applied to negative values. We get $-x \rightarrow -3x + 1$. So the standard Collatz iteration over negative values is isomorphic to iterating $x \rightarrow 3x - 1$ over positive values.
The cycles you've identified are the only known ones, and while I don't know the state of the literature I assume that if it had been proven that there were no more negative Collatz cycles then that fact would be mentioned in the Wikipedia article.
A: I don't remember longer articles about this specific variant; but if there is some notable article (before 2010, I think), then you should find it in J. Lagarias' "annotated bibliography" which I could download in various updates online over the years. I recommend the Lagarias byibliography as an entrance point.

A short overview, testing starting values initially $a_1 \le 1000$ and then later in an update by the construction-principles of the cycles and moreover for the generalization to $mx \pm1$ for $m \lt 1.14E20 $ shows a very sparse occurence of cycles in any generalization of the Collatz-problem, and confirms your finding.
For the overview see a more precise description at the header of the picture:

