We have a domino of width 1 and length $n$. We denote $A_{n}$ as the number of possible colorings of this domino, that is, the number of different ways to blacken or not blacken the cells. We denote $F_{n}$ as the number of ways to color this domino such that there are never two adjacent blackened cells.

1. Calculate $A_{n}$.
2. Calculate $F_{1}, F_{2}, F_{3}$, and $F_{4}$.
3. Find a recurrence relation between the $F_{n}$.
4. Show that $\forall n \in \mathbb{N}^{*}, \forall p \in \mathbb{N}^{*}, F_{n+p+1}=F_{n} F_{p}+F_{n-1} F_{p-1}$.

## IV. ADVANCED