On the Third-Order Jacobsthal and Third-Order Jacobsthal–Lucas Sequences and Their Matrix Representations

In this paper, we first give new generalizations for third-order Jacobsthal {Jn(3)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{J_{n}^{(3)}\}_{n\in \mathbb {N}}$$\end{document} and third-order Jacobsthal–Lucas {jn(3)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{j_{n}^{(3)}\}_{n\in \mathbb {N}}$$\end{document} sequences for Jacobsthal and Jacobsthal–Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of {Jn(3)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{J_{n}^{(3)}\}_{n\in \mathbb {N}}$$\end{document} and {jn(3)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{j_{n}^{(3)}\}_{n\in \mathbb {N}}$$\end{document}. Then, we investigate their properties.