On the number of fixed points of the map $\gamma$

We recursively define a sequence $\{F_{n,k}{\}}_{n,k\in \mathbb{N}}$ and we prove that such sequence contains only the symbols $\{0,1\}\mathrm{.}$ We investigate some number-theoretic properties of such sequence and of the way it can be generated. The number $F_n$ can be interpreted as the number of fixed points of semilength $n$ of the map $\gamma$ introduced by Barnabei et al. Our results partially answer conjectures posed by Cori et al.