Remarks on a conjecture on certain integer sequences

Abstract

The periodicity of sequences of integers ]]> ]]> ]]> ]]> ]]> ]]> ]]> (a_{n})_{n\in\mathbb Z}$ satisfying the inequalities ]]> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.