The factor complexity \(\mathcal {C}_{{\textbf {u}}}\) of a sequence \(\textbf{u}= \varvec{u}_{0}\varvec{u}_{1}\varvec{u}_{2} \cdots \) over a finite alphabet counts the number of factors of length n occurring in \(\textbf{u}\) , i.e., \(\mathcal {C}_\textbf{u}{(n)} = \#{\mathcal {L}}_n(\textbf{u})\) , where \({\mathcal {L}}_n\varvec{(}\textbf{u}\varvec{)}= {\{}\varvec{u}_{{i}}\varvec{u}_{{i+1}}\cdots \varvec{u}_{{i+n-1}}{: i} \in \mathbb {N}{\}}\) . Two factors of \({\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}\) are said to be equivalent if they are equal or one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity \({r}_\textbf{u}\) which counts the number of non-equivalent factors of \({\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}\) . They formulated the following conjecture: a sequence \(\textbf{u}\) is eventually periodic if and only if \({r}_\textbf{u}{(n+2)} = {r}_\textbf{u}{(n)}\) for some \({n} \in \mathbb {N}\) . Here we prove the conjecture and characterize the sequences for which \({r}_\textbf{u}{(n+2)} = {r}_\textbf{u}{(n)+1}\) for every \({n} \in \mathbb {N}\) and also the sequences for which the equality is satisfied for every sufficiently large \({n} \in \mathbb {N}\) .
