On the ratios of the terms of second order linear recurrences (Q1182545)

Let \(R_n\) be a nondegenerate second order linear recursive sequence of rational integers. In the case of positive discriminant the authors establish a necessary and sufficient condition for \(c>0\) and \(k>0\) such that \(|\alpha-R_{n+1}/R_n|<1/| cR_n^k|\) holds for infinitely many \(n\); \(\alpha\) denotes the characteristic root of largest modulus. In the case of negative discriminant the first author and \textit{R. F. Tichy} [Proc. Japan Acad., Ser. A 65, 135--138, 191-194 (1989; Zbl 0692.10041)] proved the existence of a positive number \(k_1<1\) such that \(\left| |\alpha| - | R_{n+1}/R_ n| \right|<1/n^{k_1}\) for infinitely many \(n\). In the present note the authors show that \(k_1\) can be replaced by 1 for the special initial values \(R_0=0\), \(R_1=1\).