Some properties of Padovan matrices and bi-periodic Padovan matrices

TL;DR

This paper investigates the diagonalizability of bi-periodic Padovan matrices, explores their connection with Lucas numbers, and relates them to the Padovan generating matrix.

Contribution

It provides new insights into the properties of bi-periodic Padovan matrices, including their diagonalizability and links to Lucas and Padovan sequences.

Findings

Conditions for diagonalizability of bi-periodic Padovan matrices

Connection between Padovan matrices and Lucas numbers

Relationships with the generating matrix for Padovan numbers

Abstract

Let $\left(P_{n}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan numbers and let $\left(M_{p_{n}}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan matrices. In this article we study when these matrices are diagonalizable and we obtain a certain connection with the Lucas number sequence. We also obtain some connections of these matrices with the generating matrix $Q$ for the Padovan numbers.