On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

Pavel Trojovský

doi:10.3390/math8081387

Abstract

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.

Highlights

The Fibonacci numbers ( Fn )n was first described in connection with computing the number of descendants of a pair of rabbits in the book Liber Abaci in 1202
This sequence is probably one of the best known recurrent sequences and it is defined by the second order recurrence
The first tool is the famous Descartes’ sign rule which gives an upper bound on the number of positive or negative real roots of a polynomial with real coefficients

Summary

Introduction

The Fibonacci numbers ( Fn )n was first described in connection with computing the number of descendants of a pair of rabbits in the book Liber Abaci in 1202 (see [1], pp. 404–405). In 2008, Włoch [30] studied the total number of k-independent sets in some special simple, undirected graphs. She showed that this number is equal to the terms of the sequence ( Tn )n≥0 , where these numbers are defined for an integer k ≥ 2 by. We are interested in the sequence ( Tn )n≥0 defined by the recurrence Equation (2), Ti but with general initial values.

Auxiliary Results

The Proof of the Main Theorem

Applications of Our Results

Conclusions