Just an amateur having fun with some sequences and would like to understand what I'm seeing. Playing with Collatz in python and found something interesting when I plotted the following:
Where I plot the trace of each path as collatz converges to $1$. Then I noticed that fewer paths plotted go into the negative $x$ direction (the first few numbers that do are: $3, 7, 9, 11, 15, 27, 31, 41, 47, 54, 55$). So I plotted the percentage of values that go into negative $x$ and how it changes as we go further into the integers and I got this plot.
This appears to going smoothly to something close to $0$? maybe. I have also been unable (through random guessing) to find a similar type of sequence that look like this. Most other attempts seem to show the number of paths that go into negative $x$ is smoothly increasing as we go through the integers.
I find myself wondering if this is something unique to collatz or if I have fooled myself into think this is property of the sequence when in reality I simply artificially created what ever patterns I'm seeing. I also find it interesting that $\frac{2n}{n/2}$ will always be $4$ and the function $\frac{3n+1}{n/3-1}$ starts behaving nicely at $4$ which is in the cycle that collatz converges to.
I'm well aware that this is probably aimless rambling but I lack the mathematical tools to prove that to myself.