TWO ASPECTS OF A GENERALIZED FIBONACCI SEQUENCE - Tuwankotta, Johan Matheus (Journal of the Indonesian Mathematical Society, 2015)
In this paper we study the so-called generalized Fibonacci sequence: $x_{n+2} = α x_{n+1} + β x_n, n\in \mathbb{N}$. We derive an open domain around the origin of the parameter space where the sequence converges to $0$. The limiting behavior on the boundary of this domain are: convergence to a nontrivial limit, $k$-periodic ($k\in \mathbb{N}$), or quasi-periodic. We use the ratio of two consecutive terms of the sequence to construct a rational approximation for algebraic numbers of the form: $\sqrt{r}, r\in \mathbb{Q}$. Using a similar idea, we extend this to higher dimension to construct a rational approximation for $\sqrt[3]{ a + b\sqrt{c}} + \sqrt[3]{ a - b\sqrt{c}} + d$.DOI : http://dx.doi.org/10.22342/jims.21.1.173.1-17