Remarks on a conjecture on certain integer sequences (Q858120)

There is the following conjecture: Let \(\lambda\in \mathbb R\) and assume that the sequence of integers \((a_n)_{n\in \mathbb Z}\) satisfies the inequalities \(0\leq a_{n-1}+\lambda a_n+a_{n+1}<1\), \(n\in \mathbb Z\). Then \((a_n)_{n\in \mathbb Z}\) is periodic for \(| \lambda| <2\). This is trivially true for \(\lambda=-1,0,1\). In this paper the conjecture is proved in the case \(\lambda=\frac{1+\sqrt{5}}{2}\), and for other values of \(\lambda\) statements on possible period lengths are given.