ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Fibonacci and Lucas Numbers

Tue 14:00–14:30, Room AV 91.21, Chair: Necati Taskara
The bi-periodic Fibonacci numbers via the determinants of special matrices
Nazmiye Yilmaz (Selcuk University)
Joint work with Necati Taskara(Selcuk University); Yasin Yazlik(Nevsehir University)

Consider the bi-periodic Fibonacci (or, equivalently, generalized Fibonacci) sequence $\left\{ q_{n}\right\} _{n=0}^{\infty }$ having initial conditions $ q_{0}=0,q_{1}=1$ and recurrence relation

\begin{equation*} q_{n}=\left\{ \begin{array}{c} aq_{n-1}+q_{n-2}, \text{if }n\text{ is even} \ bq_{n-1}+q_{n-2}, \text{if }n\text{ is odd} \end{array}% \right. n\geq 2\,, \end{equation*}

where $a$ and $b$ are nonzero real numbers [M. Edson, O. Yayenie, 2009, Integers, 9(A48)]. Some well-known sequences such as Fibonacci and Pell sequences are special cases of this generalization. In this paper, we obtain the bi-periodic Fibonacci numbers by using the determinants of some tridiagonal matrices. Also, we conclude inverses of these tridiagonal matrices.