Recursion, Catalan Numbers & Generating Functions · Challenge problem

A program computes Fibonacci numbers with the function: $\mathrm{fib}(n)$ returns $1$ if $n\le2$, and otherwise returns $\mathrm{fib}(n-1)+\mathrm{fib}(n-2)$, computing both calls from scratch. How many calls of $\mathrm{fib}$, including the first call, are made when computing $\mathrm{fib}(20)$?