We have $n$ elves and $n$ goblins. We want to line them up, but it is forbidden to place two goblins next to each other. How many possible arrangements are there?
Let's focus on the first two individuals. It is impossible to end with 2 elves since otherwise, among the $2 n-2$ last individuals, there would be $n$ goblins and $n-2$ elves, and thus there would necessarily be 2 elves next to each other (this can be easily shown by induction on $n$). The two characters are therefore either an elf followed by a goblin or a goblin followed by an elf.

We denote $v_{n}$ as the number of different arrangements that can be made with $n$ elves and $n$ goblins starting with an elf and a goblin, and $u_{n}$ as the number of arrangements possible starting with a goblin and an elf.

Thus, we have $u_{n+1}=u_{n}+v_{n}$ because if the second character is an elf, the third can be either an elf or a goblin. Similarly, we have $v_{n+1}=u_{n}$, and therefore $u_{n+1}=u_{n}+u_{n-1}$.

Since $u_{0}=u_{1}=1$, $(u_{n})$ is the Fibonacci sequence.

The number of arrangements of $n$ elves and $n$ goblins is therefore $u_{n}+v_{n}=u_{n}+u_{n-1}=u_{n+1}$.