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I'm interested in Collatz-conjecture (the 3n+1 problem) like sequences. I'm interested in any literature that contains information about problems that are divided into similar cases.

I'm particularly interested in studies of sequences that have 3 or more cases, but information on 2 case problems is interesting as well.

I'm just trying to get a feel of what is known. Anything will be helpful.

So, where can we find lots of information on these types of problems/sequences?

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For what it is worth, the people who are able to answer this question necessarily know the Collatz conjecture - stating the Collatz problem just adds noise to your question. – Thomas Andrews Jan 3 '14 at 2:08
@ThomasAndrews: I was afraid of that. I guess I was trying to keep the question self-contained so that those that don't know could learn something too. I've edited the question. – Matt Groff Jan 3 '14 at 2:48

Anyone wanting to study the $3n+1$ problem and related sequences should check Jeffrey Lagarias' annotated bibliographies on the arXiv, posted here and here. It contains some hundreds of papers, with for each paper a short summary of what it is about.

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Lagarias has also published a book in which he collects several of the important papers on Collatz. – Gerry Myerson Jan 3 '14 at 2:58

A natural generalization of the Collatz problem is recursively undecidable.

You may be interested in this paper.

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A little irritated by the language in that articled. It says, "given a Collatz function $g$, it is undecidable...." Really, what they are saying is that the general problem, not the problem for any $g$, is undecidable, right? Seems sloppy. (There are problems in the $a_i,b_i$ definitions for the original Collatz problem, too, giving me the feeling this wasn't proof-read, much less reviewed...) – Thomas Andrews Jan 3 '14 at 2:16
I agree with you. A good point. – mathlove Jan 3 '14 at 2:30
My understanding is that there exists a particular Collatz-like problem for which the usual question is undecidable. – Gerry Myerson Jan 3 '14 at 2:56
Yeah, that was my memory, too. I see the above linked article is actually a summary of a Conway article. – Thomas Andrews Jan 3 '14 at 3:07
For that Conway article, it's from the American Mathematical Monthly and you can find it at JSTOR. I'd say that's more worthwhile reading than the "paper" linked in this answer (which is not finished, and contains several sloppy mistakes). – TMM Jan 3 '14 at 14:49

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