I was going to generalize an easy counting problem and I ended up not being able to solve it:
In how many ways can $n$ people $1,\dots,n$ be seated around a round table if person $i$ refuses to sit next to person $i+1$, for all $i\in\{1,\dots,n-1\}$, and person $n$ refuses to sit next to person 1.
I realized that I have to have $n\geq5$ in order to have at least one solution.
What I have done:
it's very tempting to use the principle of inclusion and exclusion, but I'm not sure how to exactly do it, since it gets really messy very soon. Any help with this will be appreciated.
Something else that I did was using recursion, but all I'm getting is a lower bound, since I'm not able to completely analyse it:
If $n-1$ people are seated around a table in $a_{n-1}$ ways, one can place $n$ among them except for the $4$ places around $1$ and $n-1$. But then we are not counting the cases like: $\dots,2,n,3,\dots$ All it says is that $a_n \geq (n-5)a_{n-1}$.
I assume I should connect it to the case when $i$ refuses to sit next to $i+1$ for $i\in\{1,\dots,n-1\}$. But still not sure how.
It is straight forward to calculate that $a_5=2$.
PS: Is there any relation to the Menage's problem?