[seqfan] Re: Patterns of congruence classes modulo 2^n in reduced Collatz sequences

[Kleinnijenhuis, J.]

Dear Seqfans, Hi Bob Selcoe,

Reduced Collatz sequence modulo 2^n are also studied in
Thijs Laarhoven and Benne de Weger (2013), The Collatz conjecture and De Bruijn graphs. Indagationes Mathematicae 24, 971-983.

Jan Kleinnijenhuis

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Van: SeqFan [seqfan-bounces at list.seqfan.eu] namens Bob Selcoe [rselcoe at entouchonline.net]
Verzonden: vrijdag 24 april 2015 1:53
Aan: Sequence Fanatics Discussion list
Onderwerp: [seqfan] Patterns of congruence classes modulo 2^n in reduced        Collatz sequences

Hi Seqfans,

I recently found (what I think are) some interesting patterns in reduced Collatz sequences (i.e. rows in A256598).  The patterns are not obvious, but they appear to be quite regular.  Namely, the starting terms (S) can be treated as unique members of congruence classes modulo 2^n.  Membership determines how many steps it takes from S to reach the first term < its preimage (D), and how many halving steps it takes from that preimage to reach D.   There's an easy method for determining membership, and a way to streamline calculations.  The least residues are easily put into array and triangle forms for analysis.

This all may be well-understood (does any of this sound familiar??) or old hat; but even if so, I think the array is worth contributing to OEIS.  The problem is, there's a bit more information needed to describe how the residues are calculated and how to interpret them than easily fits into an OEIS entry.  I started to create the entry and ended up writing a brief outline of a paper (about 3 pages in Word) which I may be able to edit down, but it's still a bit much.  So, I was hoping there might be a way to put the outline on the OEIS somewhere, and use it as a linked reference.

Is that possible?  If so, how?  If not, are there any suggestions on how I might proceed?  Or, is anyone familiar with any papers that specifically discuss the main idea about this type of congruence relation, which might serve as a link??  I've done my best to decipher some online sources by  Jeff Lagarias and others, but I couldn't find anything on the subject (or perhaps I did and couldn't understand it!).   Anyway, I just want to contribute the entry, so any help is greatly appreciated.

Thanks in advance,
Bob Selcoe

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