(2, k)-Distance Fibonacci Polynomials

Dorota Bród,Andrzej Włoch

doi:10.3390/sym13020303

Abstract

In this paper we introduce and study (2,k)-distance Fibonacci polynomials which are natural extensions of (2,k)-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of (2,k)-distance Fibonacci polynomials with Pascal’s triangle.

Highlights

Fibonacci numbers Fn are defined by the recurrence Fn = Fn−1 + Fn−2 for n ≥ 2 with initial conditions F0 = 0, F1 = 1
Among one-parameter generalisations of Fibonacci numbers, one can find generalisations in the distance sense, such that for an arbitrary integer k the n-th generalised Fibonacci number is obtained by adding two previous terms: (n − k)-th and the second chosen in such a way that the obtained recurrence generalises Fibonacci numbers
We present connections of (2, k)-distance Fibonacci polynomials with Pascal’s 2-triangle

Summary

Introduction

Fibonacci numbers Fn are defined by the recurrence Fn = Fn−1 + Fn−2 for n ≥ 2 with initial conditions F0 = 0, F1 = 1. Let k ≥ 1 for some integer, (2, k)-distance Fibonacci polynomials we define recursively by (k). The following recurrent procedure defines the colouring c of Pn. Denote by A a set of uncoloured vertices of a path Pn. Until | A| ≤ 1, repeat following operations: Let v j ∈ A be a vertex with the smallest index. A { P2 , Pk }-scrap colouring of Pn corresponds to a sum of numbers counting vertices with the same colour, so n = n1 + n2 + ...

Matrix Generators

Connections with Pascal’s Triangle

Conclusions