I noticed that there is a linear increase of +1 in the stopping times of a sequence of numbers with the seeds being the sum of the previous number in the sequence added to itself.
I tried this on Excel. The hailstone sequences converge to 1, as expected.
I also tried it with different numbers as the starting point (ex. 5 10 20 40...)
Does this have any significance?
That is a very trivial observation. If the stopping time of a number $x$ is $n$, then the stopping time for the double of $x$: $2x$ is $n + 1$ because it takes one step for $2x$ to drop down to $x$ by division of 2.