Before I get into it, my question is what's the next step in finding the longest path for a pattern length $n$?
Imagine if you will, an integer grid. You start a bot off at $(0,0)$ with a pattern, [left, right, no turn, no turn, left, right] which the bot will then repeat forever and each step is that turn plus one unit step forward, however, if it would go to a set of coordinates it has already been to, it will instead skip that step and go to the next step, remaining in the same orientation it was in previously.
Your goal in this is to, without going infinite, make a bot that will go the furthest and then terminate. We'll shorten it to Left = $1$, Right = $-1$, and no turn/straight = $0$. To clarify, we're taking distance, not overall area.
For pattern lengths $1-3$, they're quite trivial and easy to find, so I'll leave that as an exercise for anyone interested in HOW they're found, but the longest for patterns $4-9$ are as follows:
4: $[1, 1, -1, 0]$
5: $[1, -1, 0, -1, 0]$
6: $[0, 0, 1, -1, 0, -1]$
7: $[0, 0, 0, 1, -1, 0, -1]$
8: $[0, 0, 0, 0, 1, -1, 0, -1]$
9: $[0, 1, 1, 0, 0, 0, -1, 1, 0]$
I've tried looking for a way of determining if a given pattern will stop or not, but I don't think there is one. Like $[0, 0, -1, 1, 0, 0, -1, 0, 1, 1, -1, -1]$ might stop after $3,000$ or it might not, I don't know, and I don't know enough about this type of problem to fully understand my own question.
I've tried connecting this to busy beavers, but I don't understand the math behind that either, and I've read a few papers and it's genuinely Greek to me.
I've also tried the brute force approach but I'm calculating $3x$ the number of patterns, all roughly $1.5$ times as long as the previous pattern length, so $4.5^n$. I'm also doubling back to make sure I'm checking $5$ times the length of the longest pattern I find. The way I'm determining if it does halt is if a pattern has a long length (the one I did most recently checked length $9$ patterns to $30,000$ length) I assume it doesn't halt.
There was a pattern from $6-8$ and $5$ nearly follows, but $9$ breaks it completely.
Here are what the patterns all look like:
