A Three-by-Three matrix representation of a generalized Tribonacci sequence

TL;DR

This paper introduces a generalized Tribonacci sequence defined by a specific recurrence relation and explores its properties using matrix methods, including the derivation of its generating matrix and elementary formulas.

Contribution

It presents a new generalized Tribonacci sequence and derives its matrix representation and basic properties using matrix methods, expanding understanding of third order recurrence sequences.

Findings

Derived the n-th power of the generating matrix for the sequence

Established basic properties and elementary formulas of the sequence

Provided matrix-based methods for analyzing the sequence

Abstract

The Tribonacci sequence is a well-known example of third order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Tribonacci sequence is introduced and defined by $H_{n+2}=H_{n+1}+H_{n}+H_{n-1}\ \ (n\geq 1)$, where $H_{0}=3$, $H_{1}=0$ and $H_{2}=2$. Also $n$-th power of the generating matrix for this generalized Tribonacci sequence is established and some basic properties of this sequence are obtained by matrix methods. There are many elementary formulae relating the various $H_{n}$, most of which, since the sequence is defined inductively, are themselves usually proved by induction.