I've been messing around with the Collatz sequences a bit, and have come across a few patterns – I was wondering if there are any known explanations for these patterns.
The first is the plot of stopping times (sequence lengths) for all Collatz sequences beginning with the first $100,000$ integers. As confirmed by both the wikipedia page and by this quick Python script/Excel plot, this strange line-pattern exists in the stopping times:
Another pattern was in the number of "hits" for each number. By that, I mean the number of unique sequences a specific number exists in. Generated by this similar Python script, its showing very strange patterns (if anyone wants the excel file generated, its here– just make sure you don't select the ads that look like download buttons). Specifically, one of the patterns I immediately noticed was this counting patterns – every third integer has a countdown number of hits. For example, here are the first 20 integers:
$$
\begin{array}{c|c}
1 &0 \\
2 &99998 \\
3 &16 \\
4 &99997 \\
5 &93742 \\
6 &15 \\
7 &19146 \\
8 &99996 \\
9 &14 \\
10 &93741 \\
11 &44375 \\
12 &14 \\
13 &47646 \\
14 &19145 \\
15 &13 \\
16 &99995 \\
17 &46031 \\
18 &13 \\
19 &17685 \\
20 &93724 \\
\end{array}
$$
Notice how everything else looks random, but every third integer counts down. This pattern remains throughout the whole of the $100,000$ integers (though the spaces between each different number in the countdown increase dramatically).
If anyone knows an explanation for these patterns, or can direct me towards further reading about these patterns, it'd be grealy appreciated.
Best Answer
Hmm, I'm not sure that I got you right. But it seems to me, that this is simply a consequence of any tree-structure, that there are numbers near the root through which many pathes go and that there are other numbers, far away from the root, through which less pathes go. And now, the Collatz-transformation-rule can be written in an inverted tree, say $$ \begin{array}{} & 1 \to 2 & \to 4 & \to 8 &\to 16 &\to 32 &\to \cdots\\ & & & & \to 5 &\to 10 & \to 20 & \to 40 & \to \cdots\\ & & & & & & & \to 13 & \to 26 & \to \cdots\\ & & & & & \to 3 & \to 6 & \to 12 &\to \cdots\\ \end{array} $$ or much better visible in this (very old) link then it is obvious that the number $2$ is "on the path" of many transformations, and $3$ or $13$ much less - where "many" and "less" is a bit sloppy for comparisions of infinite numbers, but it may be taken as more or less when the numbers are only taken by some finite subset.
The graphs in my link might be a bit difficult to read - I just left out the even numbers because... well, they might be seen as "trivial" and ... they didn't complain ;-)