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 .
Let , , and be integers. We define the distance -Fibonacci numbers of the second kind by the following recurrence relation: with initial conditions For we have then ; see [10]. Moreover, for and , .
In Table 2 a few first words of the distance -Fibonacci numbers of the second kind for special values of and are presented.
Let , , and be integers. We define distance -Fibonacci numbers of the third kind by the following recurrence relation: with initial conditions For we get . These numbers were introduced in [11]. For , , and we have . Moreover, for , , and .
Table 3 includes a few initial words of distance for special values of and .
By the definition of distance -Fibonacci numbers of three kinds we get for and the following relations:
3. Combinatorial and Graph Interpretations of Distance -Fibonacci Numbers
In this section we present some combinatorial and graph interpretations of distance -Fibonacci numbers. The classical Fibonacci numbers have many combinatorial interpretations. One of them is the interpretation related to set decomposition. We recall it. Let , , and be a family of disjoint subsets of such that(1),(2)if then contains two consecutive integers,(3).It is well known that the number of all families is equal to the classical Fibonacci numbers . We introduce analogous interpretation of distance -Fibonacci numbers.
Let and , , be the set of integers. Let . Assume that is a multifamily of two-element subsets of such that For fixed , by we denote a subfamily of such that . Analogously for fixed we define .
Let , , be a subfamily of such that and(a) for each , holds , for each , , holds ,(b) for each holds for and exactly one of the following conditions for and , respectively, is satisfied:(c1) or ,(c2),(c3) and if then either or .Assume that the condition (c1) is satisfied. Then the subfamily we will call a decomposition with repetitions of the set with the rest at the end.
Assume that the condition (c2) is satisfied. Then the subfamily we will call a perfect decomposition with repetitions of the set .
Assume that the condition (c3) is satisfied. Then the subfamily we will call a decomposition with repetitions of the set with the rest at the end or at the beginning.
Theorem 1. Let , , and be integers. Then the number of all decompositions with repetitions of the set with the rest at the end is equal to the number .
Proof (induction on ). Let , , and be integers. Let . Denote by the number of all decompositions with repetitions of with the rest at the end. Let . Then it is easily seen that there are exactly decompositions of . Thus we get . Let . Assume that equality holds for an arbitrary . We will show that .
Let and denote the number of all decompositions with repetitions of the set with the rest at the end such that and , respectively. It is easily seen that Moreover, we getBy the induction hypothesis and by recurrence (4) we obtainwhich ends the proof.
Analogously as Theorem 1 we can prove the following.
Theorem 2. Let , , and be integers. Then the number of all perfect decompositions with repetitions of the set is equal to the number .
Theorem 3. Let , , and be integers. Then the number of all decompositions with repetitions of the set with the rest at the end or at the beginning is equal to the number .
Distance -Fibonacci numbers of three kinds have a graph interpretation, too. It is connected with -distance -matchings in graphs. We recall the definition of a -distance -matching. Let and be any two graphs, let be an integer, and a -distance -matching of is a subgraph of such that all connected components of are isomorphic to and for each two components and from for each and holds . In case of and we obtain the definition of matching in classical sense. If covers the set (i.e., ), then we say that is a perfect matching of . For and the definition of -distance -matchings reduces to the definition of an independent set of a graph . In the literature the generalization of -matching of a graph is considered, too. For a given collection of graphs a -matching of is a family of subgraphs of such that each connected component of is isomorphic to some , . Moreover, the empty set is a -matching of , too. If for all , then we obtain the definition of -matching.
Among -matchings we consider such -matchings, where , , belong to the same class of graphs, namely, -vertex or -vertex paths ( and , resp.), .
Consider a multipath , where , , , and
Let , , and be integers. In the graph terminology the number is equal to the number of special -matchings of the multipath such that at most one vertex, namely, , does not belong to a -matching of the graph . We will call such matchings a quasi-perfect matching of . The number is equal to the number of such -matchings of that both vertex and vertex belong to some -matchings and , respectively, of the graph . In other words the number is equal to all perfect -matchings of the graph .
The number is equal to the number of special -matchings of the multipath such that at most one vertex either vertex or does not belong to a -matching of the graph .
Let be the number of all perfect -matchings of the graph .
Theorem 4. Let , , and be integers. Then .
Proof. Consider a multipath where vertices from are numbered in the natural fashion. Let and be the number of perfect -matchings of such that and , respectively. It is easily seen that .
Let be an arbitrary perfect -matching of , . Consider two cases:(1), where .
Then we can choose the edge on ways. Moreover, , where is an arbitrary -matching of the graph which is isomorphic to the multipath . Hence .(2), where .
Proving analogously as in case () we obtain .
Consequently Claim Proof. Assume now that the set corresponds to with the numbering in the natural fashion. Let be a multifamily of which gives a perfect decomposition of the set . Then every and correspond to subgraph and for , respectively, of . By Theorem 2 we get Moreover, by (6) we obtain , which ends the proof.
Analogously we can prove combinatorial interpretations of numbers and .
4. Identities for Distance -Fibonacci Numbers
In this section we give some identities and some relations between distance -Fibonacci numbers of three types.
Theorem 5. For , , and ,
Proof. We give the proof for distance -Fibonacci numbers of the first kind. By the definition of numbers , we havewhich ends the proof.
Corollary 6. For
Proof. For , , and by (19) we obtain Hence
Theorem 7. For , , and ,
Proof (induction on ). For we have Assume that equality (23) is true for an arbitrary . We will prove it for . By the recurrence (6) and by induction hypothesis we get which ends the proof.
Analogously we can prove the following.
Theorem 8. For , , and ,
Theorem 9. For , , , and ,
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to thank the referee for helpful comments and suggestions for improving an earlier version of this paper.