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Collatz sequences starting with numbers of the form $3n + 1$ for odd $n < 334$

Major Section: 
Reference
Type of Math Object: 
Example

Mathematics Subject Classification

11B37 no label found

Comments

Looking at this list of sequences, I was wondering about the
possibility of presenting the same data in tree form. For instance,
the first two sequences share the subsequence 70, 35, ..., 2, 1
in common but, in the the first sequence, this is preceded by 23
whilst, in the lattter sequence, it is preceeeded by 140. One way to
present this would be to have the subsequence 70, 3, ..., 2, 1
written in common, but to have branches 1000, 500, ..., 46, 23 and
994, 497, ..., 280, 140 preceeding it as separate branches. Likewise,
one would fit the other sequences into this tree. While they would
all agree in the ending 16,8,4,2,1, some would also coalesce much
earlier and it might be interesting to see this presented in the form
of a tree.

My opinion, for what it's worth, is that a tree presentation is the most "viable" way to present these sequences. Put 1 as the root element, from that shoot straight up a column of the powers of 2, and then from the powers of 4 branch off numbers of the form (4^x - 1)/3, and so on and so forth. I haven't actually crunched the numbers, but I have a good feeling that this approach would work perfectly for reasonably small bounds, like 1000. That's my two cents.

> Looking at this list of sequences, I was wondering
> about the possibility of presenting the same data in
> tree form.

I put a picture of part of the Collatz tree in the PM
graphics sandbox:

http://planetmath.org/encyclopedia/PlanetMathGraphicsSandbox.html

This is an excellent application for a tree diagram, and I thank Mravinci for his idea of using the powers of 2 for the central column (this had been thought of before, but rediscoverers still deserve some credit) and mps for programming the illustration using the TeX package xypic, which looks very nice. While my example object doesn't look as good, for now it has the advantage that it is searchable from the browser regardless of view. Though when someone figures out how to bestow that advantage on the tree, it will be a killer combination. Perhaps CompositeFan could suggest some idea used for The L-Word, since it is highly likely that someone has thought of using TeX to convey that show's famous chart.

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