On the third-order Jacobsthal and third-order Jacobsthal-Lucas sequences and their matrix representations

Abstract

In this paper, we first give new generalizations for third-order Jacobsthal $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and third-order Jacobsthal-Lucas $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for Jacobsthal and Jacobsthal-Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.