# Show using Fibonacci tiling ${F_0}^2 + {F_1}^2 +...+{F_n}^2 = {F_n}{F_{n+1}}$

Using double counting,

Answer 1: Consider two boards, a $(1 \times n)$-board and $(1 \times (n-1))$-board. There are ${F_{n+1}}$ ways to tile the first board and ${F_n}$ ways to tile the other. So there are ${F_n}{F_{n+1}}$ ways to tile both the boards together.

Consider the representative Fibonacci tiling below. The vertical side has height $F_n$ and the overall width is $F_n+F_{n-1}=F_{n+1}$. Thus the area shown is $F_nF_{n+1}$ which is the sum of all the squares, i.e.,
$$F_nF_{n+1}=\sum_{k=0}^nF_k^2$$
HINT: for $k,n \geq 2$, $$F_kF_n = F_{n+k-1} + F_{n-2}F_{k-2}$$