Unbounded Integer Sequences (A _n ) _n≥0 with A _n+1A _n−1− A ² _n Bounded are of Fibonacci Type

Abstract

It is well known that the Fibonacci sequence (F _n ) _{n ≥0} defined by F₀ = 0, F₁ = 1, and F _{n +2} = _{n +1} +F _n for all n ≥ 0 satisfies the condition F2 _n − F _{n +1} F _{n −1} = (-1) ⁿ for all n ∈ N In this note, we show that — somehow conversely — if (A _n ) _{n ≥0} is a sequence of integers such that (|A _n |) _{n ≥0} diverges to infinity and |A2 _n − A _{n +1} A _{n −1}|remains bounded, then (A _n ) _{n ≥0} is binary recurrent from some n on. An application to real quadratic units is also presented.