JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2015/879510 879510 Research Article On Distance (r,k)-Fibonacci Numbers and Their Combinatorial and Graph Interpretations Bród Dorota Ashrafi Ali R. Faculty of Mathematics and Applied Physics, Rzeszów University of Technology Aleja Powstańców Warszawy 12, 35-959 Rzeszów Poland prz.edu.pl 2015 1492015 2015 07 05 2015 02 09 2015 1492015 2015 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.

We introduce three new two-parameter generalizations of Fibonacci numbers. These generalizations are closely related to k-distance Fibonacci numbers introduced recently. We give combinatorial and graph interpretations of distance (r,k)-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 . The well-known Fibonacci sequence {Fn} is defined by the recurrence Fn=Fn-1+Fn-2 for n2 with F0=F1=1. 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  k-Fibonacci numbers were introduced and defined recurrently for any integer k1 by F(k,n)=kF(k,n-1)+F(k,n-2) for n2 with F(k,0)=0, F(k,1)=1. In  the following generalization of the Fibonacci numbers was defined: xn=2rxn-1+xn-2 for an integer r0 such that 4r-1+10 and n2 with x0=0 and x1=1. Other interesting generalizations of Fibonacci numbers are presented in [4, 5]. In the literature there are different kinds of distance generalizations of Fn. They have many graph interpretations closely related to the concept of k-independent sets. We recall some of such generalizations:

Reference . Consider F(k,n)=F(k,n-1)+F(k,n-k) for nk+1 with F(k,n)=n+1 for nk.

References [4, 7, 8]. Consider Fibonacci p-numbers Fp(n)=Fp(n-1)+Fp(n-p-1) for any given p(p=1,2,3,) and n>p+1 with Fp(0)=0 and Fp(n)=1 for 1np+1.

Reference . Consider Fd(1)(k,n)=Fd(1)(k,n-k+1)+Fd(1)(k,n-k) for nk with Fd(1)(k,n)=1 for nk-1.

Reference . Consider Fd(2)(k,n)=Fd(2)(k,n-k+1)+Fd(2)(k,n-k) for nk with Fd(2)(k,n)=0 for n=0,,k-2, Fd(2)(k,k-1)=1, Fd(2)(1,1)=1, Fd(2)(2,2)=2, for k3Fd(2)(k,k)=1.

Reference . Consider Fd(3)(k,n)=Fd(3)(k,n-k+1)+Fd(3)(k,n-k) for n2k-1 with Fd(3)(k,n)=1 for n=0,,k-1, Fd(3)(2,2)=2, for k3Fd(3)(k,k)=Fd(3)(k,2k-2)=3, for k+1n2k-1Fd(3)(k,n)=4.

Reference . Consider F2(1)(k,n)=F2(1)(k,n-2)+F2(1)(k,n-k) for nk+1 with (1)F21k,n=1ifnk-1  or  n=k=1,2ifn=k2.

Reference . Consider F2(2)(k,n)=F2(2)(k,n-2)+F2(2)(k,n-k) for nk+1 with(2)F22k,n=0if  n  is  odd  andnk-1,1if  n  is  even  andnk-1,F22k,k=0if  k=1,1if  k  is  odd  and  k3,2if  k  is  even.

Reference . Consider F2(3)(k,n)=F2(3)(k,n-2)+F2(3)(k,n-k) for nk+1 with(3)F23k,n=1if  n  is  even  and  nk-1,2if  n  is  odd  and  nk-1,F23k,k=3if  k  is  odd  and  k3,2if  k  is  even  or  k=1.

In this paper we introduce three new two-parameter generalizations of distance Fibonacci numbers. They are closely related with the numbers F2(j)(k,n), j=1,2,3, presented in [10, 11]. We show their combinatorial and graph interpretations and we present some identities for them.

2. Distance <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M63"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>r</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Fibonacci Numbers

Let k1, n0, and r1 be integers. We define distance (r,k)-Fibonacci numbers of the first kind Fr(I)(k,n) by the recurrence relation (4)FrIk,n=rFrIk,n-2+rk-1FrIk,n-kfornk+1with the following initial conditions:(5)FrIk,0=FrIk,1=1,FrIk,n=rn/2forn=2,3,,k-2,FrIk,k-1=rk-1/2fork3,FrIk,k=rk-1+rk/2fork2.For r=1 we get F1(I)(k,n)=F2(1)(k,n). These numbers were introduced in .

If r=1 and k=1, then F1(I)(1,n) gives the Fibonacci numbers Fn. For r=1 and k=3 the numbers F1(I)(3,n) are the well-known Padovan numbers.

Table 1 includes the values of Fr(I)(k,n) for special values of k and n.

Distance (r,k)-Fibonacci numbers Fr(I)(k,n) of the first kind.

k n 0 1 2 3 4 5 6 7 8
1 1 1 r + 1 2 r + 1 r 2 + 3 r + 1 r 2 + 5 r + 2 r 3 + 4 r 2 + 6 r + 2 2 r 3 + 9 r 2 + 8 r + 2 r 4 + 6 r 3 + 15 r 2 + 10 r + 2
2 1 1 2 r 2 r 4 r 2 4 r 2 8 r 3 8 r 3 16 r 4
3 1 1 r r 2 + r 2 r 2 2 r 3 + r 2 r 4 + 3 r 3 4 r 4 + r 3 3 r 5 + 4 r 4
4 1 1 r r r 3 + r 2 r 3 + r 2 2 r 4 + r 3 2 r 4 + r 3 r 6 + 3 r 5 + r 4
5 1 1 r r r 2 r 4 + r 2 r 4 + r 3 2 r 5 + r 3 2 r 5 + r 4
6 1 1 r r r 2 r 2 r 5 + r 3 r 5 + r 3 2 r 6 + r 4
7 1 1 r r r 2 r 2 r 3 r 6 + r 3 r 6 + r 4

Let k1, n0, and r1 be integers. We define the distance (r,k)-Fibonacci numbers of the second kind Fr(II)(k,n) by the following recurrence relation: (6)FrIIk,n=rFrIIk,n-2+rk-1FrIIk,n-kfornk+1 with initial conditions(7)FrIIk,n=rn/2for  even  n,0for  odd  n,forn=0,1,,k-1FrIIk,k=0for  k=1,rk-1for  oddk,k3,rk-1+rk/2for  even  k. For r=1 we have then F1(II)(k,n)=F2(2)(k,n); see . Moreover, for r=1 and k=1, n2F1(II)(k,n)=Fn-2.

In Table 2 a few first words of the distance (r,k)-Fibonacci numbers of the second kind Fr(II)(k,n) for special values of k and n are presented.

Distance (r,k)-Fibonacci numbers Fr(II)(k,n) of the second kind.

k n 0 1 2 3 4 5 6 7 8
1 1 0 r r r 2 + r 2 r 2 + r r 3 + 3 r 2 + r 3 r 3 + 4 r 2 + r r 4 + 6 r 3 + 5 r 2 + r
2 1 0 2 r 0 4 r 2 0 8 r 3 0 16 r 4
3 1 0 r r 2 r 2 2 r 3 r 4 + r 3 3 r 4 3 r 5 + r 4
4 1 0 r 0 r 3 + r 2 0 2 r 4 + r 3 0 r 6 + 3 r 5 + r 4
5 1 0 r 0 r 2 r 4 r 3 2 r 5 r 4
6 1 0 r 0 r 2 0 r 5 + r 3 0 2 r 6 + r 4
7 1 0 r 0 r 2 0 r 3 r 6 r 4

Let k1, n0, and r1 be integers. We define distance (r,k)-Fibonacci numbers of the third kind Fr(III)(k,n) by the following recurrence relation: (8)FrIIIk,n=rFrIIIk,n-2+rk-1FrIIIk,n-kfor  nk+1 with initial conditions (9)FrIII1,1=2,FrIIIk,n=rn/2for  even  n,2rn/2for  odd  n,for  n=0,1,,k-1FrIIIk,k=rk-1+rk/2for  even  k,rk-1+2rk/2for  odd  k3.For r=1 we get F1(III)(k,n)=F2(3)(k,n). These numbers were introduced in . For r=1, k=1, and n0 we have F1(III)(1,n)=Fn+1. Moreover, for r=1, k=4, and n1F1(III)(4,2n)=Fn.

Table 3 includes a few initial words of distance Fr(III)(k,n) for special values of k and n.

Distance (r,k)-Fibonacci numbers Fr(III)(k,n) of the third kind.

k n 0 1 2 3 4 5 6 7 8
1 1 2 r + 2 3 r + 2 r 2 + 5 r + 2 4 r 2 + 7 r + 2 r 3 + 9 r 2 + 9 r + 2 5 r 3 + 9 r 2 + 16 r + 4 r 4 + 14 r 3 + 18 r 2 + 18 r + 4
2 1 2 2 r 4 r 4 r 2 8 r 2 8 r 3 16 r 3 16 r 4
3 1 2 r r 2 + 2 r 3 r 2 2 r 3 + 2 r 2 r 4 + 5 r 3 5 r 4 + 2 r 3 3 r 5 + 7 r 4
4 1 2 r 2 r r 3 + r 2 2 r 3 + 2 r 2 2 r 4 + r 3 4 r 4 + 2 r 3 r 6 + 3 r 5 + r 4
5 1 2 r 2 r r 2 r 4 + 2 r 2 2 r 4 + r 3 2 r 5 + 2 r 3 4 r 5 + r 4
6 1 2 r 2 r r 2 2 r 2 r 5 + r 3 2 r 5 + 2 r 3 2 r 6 + r 4
7 1 2 r 2 r r 2 2 r 2 r 3 r 6 + 2 r 3 2 r 6 + 2 r 4

By the definition of distance (r,k)-Fibonacci numbers of three kinds we get for k1 and n0 the following relations:(10)FrIIIk,n=2FrIk,nfor  even  k  and  odd  n,FrIk,n=FrIIk,n=FrIIIk,nfor  even  k  and  even  n,FrIIk,n=0for  even  k  and  odd  n.

3. Combinatorial and Graph Interpretations of Distance <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M368"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>r</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Fibonacci Numbers

In this section we present some combinatorial and graph interpretations of distance (r,k)-Fibonacci numbers. The classical Fibonacci numbers have many combinatorial interpretations. One of them is the interpretation related to set decomposition. We recall it. Let X={1,2,,n}, n1, and Y={Yt:tT} be a family of disjoint subsets of X such that

Yt{1,2},

if Yt=2 then Yt contains two consecutive integers,

XtTYt=.

It is well known that the number of all families Y is equal to the classical Fibonacci numbers Fn. We introduce analogous interpretation of distance (r,k)-Fibonacci numbers.

Let r1 and X={1,2,,n}, n2, be the set of n integers. Let k3. Assume that Rn is a multifamily of two-element subsets of X such that (11)Rn=1,2,1,2,,1,2r-times,2,3,,2,3r-times,,n-1,n,,n-1,nr-times.For fixed t, 1tn-k by R(k,t) we denote a subfamily of Rn such that R(k,t)={{t+j,t+j+1}:j=0,1,,k-2,t=1,2,,n-k+1}. Analogously for fixed t we define R(k,t)={{t,t+1}:t=1,2,,n-1}.

Let Rt,t(j), j=1,2,3, be a subfamily of Rn such that Rt,t(j)=R(k,t)R(k,t) and

for each R(k,t1),R(k,t2)R(k,t), t1t2 holds t1-t22, for each R(k,t1),R(k,t2)R(k,t), t1t2, holds t1-t2k,

for each RtRt,t(j) holds Rt{2,k} for tT

and exactly one of the following conditions for RtRt,t(j) and j=I,II,III, respectively, is satisfied:

XtTRt= or XtTRt=n,

XtTRt=,

XtTRt{0,1} and if pXtTRt then either p=1 or p=n.

Assume that the condition (c1) is satisfied. Then the subfamily Rt,t(1) we will call a decomposition with repetitions of the set X with the rest at the end.

Assume that the condition (c2) is satisfied. Then the subfamily Rt,t(2) we will call a perfect decomposition with repetitions of the set X.

Assume that the condition (c3) is satisfied. Then the subfamily Rt,t(3) we will call a decomposition with repetitions of the set X with the rest at the end or at the beginning.

Theorem 1.

Let k3, n2, and r1 be integers. Then the number of all decompositions with repetitions of the set X with the rest at the end is equal to the number Fr(I)(k,n).

Proof (induction on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M429"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>).

Let k3, n2, and r1 be integers. Let X={1,2,,n}. Denote by d(n) the number of all decompositions with repetitions of X with the rest at the end. Let n=2. Then it is easily seen that there are exactly r decompositions of X. Thus we get d(2)=r=Fr(I)(k,2). Let n3. Assume that equality d(n)=Fr(I)(k,n) holds for an arbitrary n. We will show that d(n+1)=Fr(I)(k,n+1).

Let d2(n+1) and dk(n+1) denote the number of all decompositions R with repetitions of the set X={1,2,,n+1} with the rest at the end such that {1,2}R and {1,2,,k}R, respectively. It is easily seen that (12)dn+1=d2n+1+dkn+1. Moreover, we get(13)d2n+1=dkn-1,dkn+1=dkn+1-k.By the induction hypothesis and by recurrence (4) we obtain(14)dn+1=dn-1+dn+1-k=FrIk,n-1+FrIk,n+1-k=FrIk,n+1,which ends the proof.

Analogously as Theorem 1 we can prove the following.

Theorem 2.

Let k3, n2, and r1 be integers. Then the number of all perfect decompositions with repetitions of the set X is equal to the number Fr(II)(k,n).

Theorem 3.

Let k3, n2, and r1 be integers. Then the number of all decompositions with repetitions of the set X with the rest at the end or at the beginning is equal to the number Fr(III)(k,n).

Distance (r,k)-Fibonacci numbers of three kinds have a graph interpretation, too. It is connected with k-distance H-matchings in graphs. We recall the definition of a k-distance H-matching. Let G and H be any two graphs, let k1 be an integer, and a k-distance H-matching M of G is a subgraph of G such that all connected components of M are isomorphic to H and for each two components H1 and H2 from M for each xV(H1) and yV(H2) holds dG(x,y)k. In case of k=1 and H=K2 we obtain the definition of matching in classical sense. If M covers the set V(G) (i.e., V(M)=V(G)), then we say that M is a perfect matching of G. For k=2 and H=K1 the definition of k-distance H-matchings reduces to the definition of an independent set of a graph G. In the literature the generalization of H-matching of a graph G is considered, too. For a given collection H=H1,H2,,Hn of graphs a H-matching M of G is a family of subgraphs of G such that each connected component of M is isomorphic to some Hi, 1in. Moreover, the empty set is a H-matching of G, too. If Hi=H for all i=1,2,,n, then we obtain the definition of H-matching.

Among H-matchings we consider such H-matchings, where Hi, i=1,2,,n, belong to the same class of graphs, namely, 2-vertex or k-vertex paths (P2 and Pk, resp.), k3.

Consider a multipath Pnr, where n2, r1, V(Pnr)={x1,x2,,xn}, and(15)EPnr=x1,x2,,x1,x2r-times,x2,x3,,x2,x3r-times,,xn-1,xn,,xn-1,xnr-times.

Let n2, k3, and r1 be integers. In the graph terminology the number Fr(I)(k,n) is equal to the number of special {P2,Pk}-matchings M of the multipath Pnr such that at most one vertex, namely, xn, does not belong to a {P2,Pk}-matching of the graph Pnr. We will call such matchings M a quasi-perfect matching of Pnr. The number Fr(II)(k,n) is equal to the number of such {P2,Pk}-matchings of Pnr that both vertex x1 and vertex xn belong to some {P2,Pk}-matchings M and M, respectively, of the graph Pnr. In other words the number Fr(II)(k,n) is equal to all perfect {P2,Pk}-matchings M of the graph Pnr.

The number Fr(III)(k,n) is equal to the number of special {Pk,P2}-matchings of the multipath Pnr such that at most one vertex either vertex x1 or xn does not belong to a {P2,Pk}-matching of the graph Pnr.

Let σ(Pnr) be the number of all perfect {P2,Pk}-matchings M of the graph Pnr.

Theorem 4.

Let r1, k3, and n2 be integers. Then σ(Pnr)=Fr(II)(k,n).

Proof.

Consider a multipath Pnr where vertices from V(Pnr)={x1,x2,,xn} are numbered in the natural fashion. Let σk(n) and σ2(n) be the number of perfect {P2,Pk}-matchings M of Pnr such that xn,xn-1V(M) and xn,xn-1,,xn-kV(M), respectively. It is easily seen that σk(n)+σ2(n)=σ(Pnr).

Let M be an arbitrary perfect {P2,Pk}-matching of Pnr, k3. Consider two cases:

{xn-1,xn}E(Pk), where PkM.

Then we can choose the edge {xn-1,xn} on r ways. Moreover, M=M{Pk}, where M is an arbitrary {P2,Pk}-matching of the graph Pnr{xn,xn-1,,xn-k+1} which is isomorphic to the multipath Pn-kr. Hence σk(n)=rk-1σ(Pn-kr).

{xn-1,xn}E(P2), where P2M.

Proving analogously as in case (1) we obtain σ2(n)=rσ(Pn-2r).

Consequently (16)σPnr=σkn+σ2n=rk-1σPn-kr+rσPn-2r.Claim (17)σPnr=rk-1FrIIk,n-k+rFrIIk,n-2.

Proof. Assume now that the set X={1,2,,n} corresponds to V(Pnr) with the numbering in the natural fashion. Let R(t,t)={Rt:tT}{Rt:tT} be a multifamily of X which gives a perfect decomposition of the set X. Then every Rt and Rt correspond to subgraph P|Rt| and P|Rt| for t,tT, respectively, of Pnr. By Theorem 2 we get (18)σPnr=σkn+σ2n=rk-1FrIIk,n-k+rFrIIk,n-2. Moreover, by (6) we obtain σ(Pnr)=Fr(II)(k,n), which ends the proof.

Analogously we can prove combinatorial interpretations of numbers Fr(I)(k,n) and Fr(III)(k,n).

4. Identities for Distance <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M610"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>r</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Fibonacci Numbers

In this section we give some identities and some relations between distance (r,k)-Fibonacci numbers of three types.

Theorem 5.

For k1, n2k-2, and j=I,II,III, (19)Frjk,n=rFrjk,n-2+rk-2Frjk,n-k+2-r2k-3Frjk,n-2k+2.

Proof.

We give the proof for distance (r,k)-Fibonacci numbers of the first kind. By the definition of numbers Fr(I)(k,n), we have(20)rFrIk,n-2+rk-2FrIk,n-k+2-r2k-3FrIk,n-2k+2=rFrIk,n-2+rk-2rFrIk,n-k+rk-1FrIk,n-2k+2-r2k-3FrIk,n-2k+2=rFrIk,n-2+rk-1F2Ik,n-k=FrIk,n,which ends the proof.

Corollary 6.

For n2Fn=1/2(Fn-2+Fn+1).

Proof.

For r=1, j=I, and k=1 by (19) we obtain (21)F1I1,n=Fn=Fn-2+Fn+1-Fn.Hence (22)Fn=12Fn-2+Fn+1.

Theorem 7.

For r1, k2, and n1, (23)FrIk,n=FrIIk,n+FrIIk,n-1.

Proof (induction on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M629"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>).

For n=1 we have (24)FrIk,1=1=FrIIk,1+FrIIk,0. Assume that equality (23) is true for an arbitrary n. We will prove it for n+1. By the recurrence (6) and by induction hypothesis we get (25)FrIk,n+1=rFrIk,n-1+rk-1FrIk,n+1-k=rFrIIk,n-1+FrIIk,n-2+rk-1FrIIk,n+1-k+FrIIk,n-k=rFrIIk,n-1+rk-1FrIIk,n+1-k+rFrIIk,n-2+rk-1F2IIk,n-k=FrIIk,n+1+FrIIk,n, which ends the proof.

Analogously we can prove the following.

Theorem 8.

For r1, k3, and n0, (26)2FrIk,n=FrIIk,n+FrIIIk,n.

Theorem 9.

For r1, k2, n2k, and j=I,II,III, (27)Frjk,n=r2Frjk,n-4+2rkFrjk,n-k-2+r2k-2Frjk,n-2k.

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.

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