Main loop initializes $= each time to 1, and creates a string of sizes 0, 1, 2, ... 10. This string is matched against the regexp.
Note that the pattern does not match (due to the trailing ^), so basically what's happening that $= counts the number of times the regexp engine will backtrack. We can simplify the pattern a little:
/^(1|11)*^/
(the capture and backref of the anchor are just there to confuse the optimizer). So, what the pattern is counting how many times it can partition a string into 1 and 2 character substrings. Let's call this number
F(n), for a string of size
n. The pattern says "either match '1' or match '11'; then recurse with the rest". How many times can we do that?
F(n+1) + F(n+2). So, we derive:
F(n) = F(n+1) + F(n+2)
Hmmm. That's exactly the definition of the Fibonacci series (except for the initial condition).