You are climbing a staircase. At each step, you can either make $1$ step climb, or make $2$ steps climb. Say a staircase of height of $3$. You can climb in $3$ ways $(1-1-1,\ 1-2,\ 2-1)$.
Say a staircase of height of $4$, You can climb in $5$ ways.
Given a staircase of height of $n$, can you figure out how many ways you can climb?
This is actually a programming problem, I have already written the C++ code in recursion, but I just don't know how to verify my program using mathematical skills. I feel this is not a complicate math problem, but yet I couldn't solve it. So I am asking for your help.