Cayley unitary elements in group algebras under oriented involutions

Let $\mathbf{F}$ be a real extension of $\mathbb{Q}$, $G$ a finite group and $\mathbf{F}G$ its group algebra. Given both a group homomorphism $\sigma:G\rightarrow \{\pm1\}$ (called an orientation) and a group involution $^\ast:G \rightarrow G$ such that $gg^\ast\in N=ker(\sigma)$, an oriented group involution $\circledast$ of $\mathbf{F}G$ is defined by $\alpha=\sum_{g\in G}\alpha_{g}g \mapsto \alpha^\circledast=\sum_{g\in G}\alpha_{g}\sigma(g)g^{\ast}$. In this paper, in case the involution on $G$ is the classical one, $x\mapsto x^{-1}$, $\beta=x+x^{-1}$ is a skew-symmetric element in $\mathbf{F}G$ such that $1+\beta$ is invertible, for $x\in G$ with $\sigma(x)=-1$, we consider Cayley unitary elements built out of $\beta$. We prove that the coefficients of $(1+\beta)^{-1}$ involve an interesting sequence which is a Fibonacci-like sequence.