About a combinatorial problem with $n$ seats and $n$ people

If you want to fill $n\in \mathbb{N}$ seats in succession with $n$ people and the rule that each person chooses one of the seats with the maximum distance to an occupied seat, then you can ask yourself how many possibilities there are for this. In this paper, based on initially mentioned ideas, a formula for the number of these possibilities will be found. In addition, a lower and upper bound for this formula will be given. Finally, formulas for the OEIS sequences A166079, A095236, A095240 and A095912 and an extension of the initial problem are derived.