Journal of Applied Mathematics

Volume 2015, Article ID 879510, 6 pages

http://dx.doi.org/10.1155/2015/879510

## On Distance -Fibonacci Numbers and Their Combinatorial and Graph Interpretations

Faculty of Mathematics and Applied Physics, Rzeszów University of Technology, Aleja Powstańców Warszawy 12, 35-959 Rzeszów, Poland

Received 7 May 2015; Accepted 2 September 2015

Academic Editor: Ali R. Ashrafi

Copyright © 2015 Dorota Bród. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce three new two-parameter generalizations of Fibonacci numbers. These generalizations are closely related to -distance Fibonacci numbers introduced recently. We give combinatorial and graph interpretations of distance -Fibonacci numbers. We also study some properties of these numbers.

#### 1. Introduction

In general we use the standard terminology of the combinatorics and graph theory; see [1]. The well-known Fibonacci sequence is defined by the recurrence for with . The Fibonacci numbers have been generalized in many ways, some by preserving the initial conditions and others by preserving the recurrence relation. For example, in [2] -Fibonacci numbers were introduced and defined recurrently for any integer by for with , . In [3] the following generalization of the Fibonacci numbers was defined: for an integer such that and with and . Other interesting generalizations of Fibonacci numbers are presented in [4, 5]. In the literature there are different kinds of distance generalizations of . They have many graph interpretations closely related to the concept of -independent sets. We recall some of such generalizations:(1)Reference [6]. Consider for with for .(2)References [4, 7, 8]. Consider Fibonacci -numbers for any given and with and for .(3)Reference [9]. Consider for with for .(4)Reference [9]. Consider for with for , , , , for (5)Reference [9]. Consider for with for , , for , for (6)Reference [10]. Consider for with (7)Reference [11]. Consider for with(8)Reference [11]. Consider for with

In this paper we introduce three new two-parameter generalizations of distance Fibonacci numbers. They are closely related with the numbers , , presented in [10, 11]. We show their combinatorial and graph interpretations and we present some identities for them.

#### 2. Distance -Fibonacci Numbers

Let , , and be integers. We define distance -Fibonacci numbers of the first kind by the recurrence relation with the following initial conditions:For we get . These numbers were introduced in [10].

If and , then gives the Fibonacci numbers . For and the numbers are the well-known Padovan numbers.

Table 1 includes the values of for special values of and .