Distance Fibonacci Numbers, Distance Lucas Numbers and Their Applications

Abstract

In this paper we define new generalizations of the Lucas numbers,which also generalize the Perrin numbers. This generalization is based on the concept of \(k\)-distance Fibonacci numbers. We give in-terpretations of these numbers with respect to special decompositions and coverings, also in graphs. Moreover, we show some identities for these numbers, which often generalize known classical relations for the Lucas numbers and the Perrin numbers. We give an application of the distance Fibonacci numbers for building the Pascal’s triangle.