Fibonacci numbers, arctangents, and pi

Here's an unusual formula for I. Let F_n be the nth Fibonacci number. Then

As mysterious as this equation may seem, it's not hard to prove. The arctangent identity

shows that the sum telescopes, leaving only the first term, arctan(1) = I/4. To prove the arctangent identity, take the tangent of both sides, use the addition law for tangents, and use the Fibonacci identity

See this post for an even more remarkable formula relating Fibonacci numbers and I.