I made a "discovery" a couple of weeks ago in regards to the first iterates of (odd) numbers of the form $2^n-1$ where $n\in\mathbb{N}$. First iterates is a bit loose term here; what I mean is all of the first consequtive odd numbers of the Collatz function iterates. The similarities are shown in the space-time diagrams of Rule $150$ and the first consequtive odd collatz iterates:
Iterates of the Collatz function: $(3n+1)/2$ if odd, or $n/2$ if even.
Rule 150 Elementary Cellular Automaton. Image courtesy WolframAlpha.
I have written a more detailed description of this discovery on my blog: https://sites.google.com/site/rudibstranden/
Since this happens for just a small subset of the domain it is still a surprising phenomenom (for me atleast). What I also pointed out in my blog is that the type of randomness or intricate structures that we see (the stuff inside pyramid) is produced with no random initial condition of the first input (since its just a string of $1$'s), but in Rule 150 this is total opposite: there is a random initial-condition (random $1$'s and $0$'s) that produce these structures as seen in the second figure above.
People who study Cellular Automata knows that this particular Rule (Rule $150$) is not often seen for Elementary Cellular Automata, like other rules ($60$, $90$ and so on) this is only happens once or twice out of the 256 rules (that has reversible or injective features). Does somebody knows what happens for these kinds of inputs and why it looks similar to Rule 150? I am also looking for a paper that explains why these structures look so similar and why this behaviour happens? Or I am wondering if I should write a paper about it, since I have not found this anywhere before. Thanks.
Edit: What kind of analytical tools (and analysis) can one use on these kinds of structures?