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.