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)*^/

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(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)

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Hmmm. That's exactly the definition of the Fibonacci series (except for the initial condition).

In reply to Re: Explain Fibonacci one-liner ?? by JavaFan
in thread Explain Fibonacci one-liner ?? by Anonymous Monk

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