# On the third-order Horadam matrix sequences

## TL;DR

This paper introduces new generalizations of third-order Horadam and Tribonacci sequences, defines related matrix sequences, and explores their properties, expanding the mathematical understanding of these recursive sequences.

## Contribution

The paper presents novel generalizations of third-order Horadam and Tribonacci sequences and analyzes the properties of their associated matrix sequences.

## Key findings

- New generalizations for third-order Horadam and Tribonacci sequences
- Definition and analysis of matrix sequences based on these generalized sequences
- Investigation of properties of the new matrix sequences

## Abstract

In this paper, we first give new generalizations for third-order Horadam $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and generalized Tribonacci $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for classic Horadam and generalized Fibonacci numbers. Considering these sequences, we define the matrix sequences which have elements of $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.