@SimonHenry looking more closely at one of my proofs I just realized that this is exactly the 3n+1 problem. That is to say, any sequence satisfying this equation for an odd s exactly corresponds to meeting the condition I mentioned earlier. Sorry, had to go back and read my paper slowly. I did not realize this earlier.

@GottfriedHelms thank you for the reference! Your notation and mine however are not equivalent. Also, the ideas behind the above formula are very much different from your work.

@GottfriedHelms your first observation is correct. It’s hard to get any insight out of that if any though. As for your second observation, could you please provide the wikipedia link and section? The closest thing I found was a reference to rational representations which is not quite this, and not even an equation at that. The condition takes a bit to write down, but it remains an open problem to me whether the only solutions to this are exactly those that meet those restrictions; in that case, this problem is exactly equivalent to the collatz conjecture. I am not sure I could prove the link.

The 3n+1 problem has a condition on $a_n$ which is not required here.

Thanks!! I like this solution better, but I appreciate the variety of perspectives to solve this.