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Some properties of Padovan matrices and bi-periodic Padovan matrices

Diana Savin

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.

Some properties of Padovan matrices and bi-periodic Padovan matrices

Abstract

Let be the sequence of bi-periodic Padovan numbers and let 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 for the Padovan numbers.

Paper Structure

This paper contains 2 sections, 7 theorems, 39 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Results

Key Result

Proposition 2.1

Let $\left(M_{p_{n}}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan matrices defined in the introduction section. Then $M_{p_{n}}\cdot M_{p_{n+1}}= M_{p_{n+1}}\cdot M_{p_{n}}, \ \left(\forall\right) \ n\geq 0.$

Theorems & Definitions (9)

  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9