Consider the problem of tiling a board of length $n$ with squares of size $1×1$ and dominoes of size $1×2$, Let's denote $f_n$ to be the number of ways to tile this so-called ($n$)-board.Then $f_n=F_{n+1}$, where $F_n$ is the nth Fibonacci number.
Now the question is as follows:
Is there any combinatorial way to prove Catalan's identity which states: $$F_{n}^2-F_{n-r}F_{n+r}=(-1)^{n-r}F_r^2$$
It's easy to prove this identity using induction or using a more generalized form of that which is indeed Vajda's identity.
Cassini's identity (a special case of Catalan's identity ) has been proved using combinatorial interpretation in a few books, but I've never seen any combinatorial proof of this one.