A Characterization of Certain Binary Recurrence Sequences

In this note, we show that if (A_n) _n≥0 is a sequence of integers such that (|A_n|) _n≥0 diverges to infinity and

$$\mathop{{\lim \sup }}\limits_{{n \to \infty }} \frac{{|A_{n}^{2} - {{A}_{{n + 1}}}{{A}_{{n - 1}}}|}}{{\sqrt {{|{{A}_{n}}|}} }} < \frac{1}{{\sqrt {2} }}$$

holds, then (A_n)_n≥0 is binary recurrent from some n on. An application regarding quadratic Pisot numbers is also presented.