**Contributed talks on Fibonacci and Lucas Numbers**

The bi-periodic Fibonacci numbers via the determinants of special matrices

Nazmiye Yilmaz (Selcuk 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.