I'm not sure how related this particular example is to your question (but I think it's interesting). generatingfunctionology contains a proof that the Fibonacci numbers are approximated by \[F_n \sim \frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n.\] It is followed by the remark that the error term is never more then 1/2. Hence F_n is the nearest integer to the above approximation.