As the title states, I am tasked with finding finding a recurrence relation according to the specific constraints. I have a start to the problem, but I'm really having trouble identifying the pattern for the recurrence relation. My first step was to go through the first few cases to see if a pattern emerged. Here is what I found:
Here's where I got a bit stuck. I am currently trying to look for patterns in terms of taking the previous terms and adding a combination of 1's and 0's to them. I found the terms for length 5 by taking the previous term (length 4) and adding a 1 to the front of all the terms, then adding a 0 to the front of all the terms, then a 1 to the back, then a 0 to the back. I ended up with a few duplicates that I then had to remove. Any help towards the next steps I need to take or hints would be greatly appreciated!