The Binet-Like Formula of a Family of Conditional Sequences by Matrix Methods

Abstract

Let $a_0,a_1,\dots ,a_{r-1}$ be positive integers and define a conditional sequence $\{q_n\}$, with initial conditions $q_0=0$ and $q_1=1$, and for all $n\ge 2$, $q_n=a_1{q}_{n-1}+q_{n-2}$ where $n\equiv t(\mathrm{mod}r)$. For $r=2$, the author studied it in $[1]$. For general $\{q_n\}$, we found a closed form of the generating function for $\{q_n\}$ in terms of the continuant in $[2]$. In this paper, we give the matrix representation and a Binet-like formula for the conditional sequence $\{q_n\}$ by using the matrix methods.