  We recursively define a sequence $\{F_{n,k}\}_{n,k\in\mathbb N }$ and we
prove that such sequence contains only the symbols $\{0,1\}.$ 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.
