Fibonacci Quart Research Articles

On degenerate multi-poly-derangement polynomials and numbers

ABSTRACT The problem of counting derangements was begun in 1708 by Pierre R e ´ mond de Montmort (see [Carlitz. The number of derangements of a sequence with given specification. Fibonacci Quart. 1978;16:255–258], [Clarke and Sved. Derangements and Bell numbers. Math Mag. 1993;66:299–303]), and its applications have been investigated actively by many researchers. In this paper, we study properties of the degenerate multi-poly-derangement polynomials and numbers that are defined by using the multiple logarithm function, and derive some interesting identities related to these polynomials and numbers.

Combinatorial proof of Girard-Waring formula

\noindent The well-known and celebrated identity of Girard (1629) and Waring (1762) states that \[x^n+y^n=\sum_{m=0}^{\lfloor n/2\rfloor}(-1)^m\frac{n}{n-m}\binom{n-m}{m}\left(xy\right)^m\left(x+y\right)^{n-2m}\] and can be easily proven algebraically (see H.W. Gould, The Girard--Waring power sum formulas for symmetric functions and Fibonacci sequences,} Fibonacci Quart. 37 (1999), no. 2, 135--140). In this note, we provide a combinatorial proof of this identity.

ALTERNATING CIRCULAR SUMS OF BINOMIAL COEFFICIENTS

AbstractBy combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart.3(2) (1965), 81–89].

On $k$-Fibonacci balancing and $k$-Fibonacci Lucas-balancing numbers

The balancing number $n$ and the balancer $r$ are solution of the Diophantine equation $$1+2+\cdots+(n-1) = (n+1)+(n+2)+\cdots+(n+r). $$ It is well known that if $n$ is balancing number, then $8n^2 + 1$ is a perfect square and its positive square root is called a Lucas-balancing number. For an integer $k\geq 2$, let $(F_n^{(k)})_n$ be the $k$-generalized Fibonacci sequence which starts with $0,\ldots,0,1,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of $k$-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].

Degenerate central Bell numbers and polynomials

Here we study the degenerate central Bell numbers and polynomials as a degenerate version of the recently introduced central Bell numbers and polynomials, which are motivated by Zhang’s work ‘Some identities involving the Euler and the central factorial numbers’ (Zhang in Fibonacci Quart 36(2):154–157, 1998) and we derive some properties, identities, and recurrence relations for these numbers and polynomials. In particular, we find various expressions for the degenerate central Bell numbers and polynomials.

Ducci sequences of triples and continued fractions

Using continued fraction expansions, we provide an explicit expression for the orbit of a real vector under the Ducci map. This expression enables us to give an alternative proof of Brockman and Zerr’s characterization theorem on the behaviour of the Ducci sequences on [Fibonacci Quart. 45 (2007), pp. 155–163]. We then consider Ducci matrix sequences, a concept introduced by Hogenson et al. [Linear Algebra Appl. 437 (2012), pp. 285–293], and provide a new proof of their main theorem. Moreover, utilizing the explicit expression, we compute, for two given distinct starting vectors in , the length of a maximal common initial segment of their Ducci matrix sequences. Finally, answering a question by Hogenson et al., we prove that for a given positive irrational number , the Ducci matrix sequence associated with is eventually periodic if and only if is quadratic.

Grundy values of Fibonacci nim

In this article, we investigate the Grundy values of the popular game of Fibonacci nim. The winning strategy, which amounts to understanding positions of Grundy value 0, was known since Whinihan (Fibonacci Quart 1:9–13, 1963). In this paper, we extend Whinihan’s analysis by computing all the positions of Grundy value at most 3. Furthermore, we show that, when we delete the Fibonacci numbers (which have Grundy value 0), the Grundy values of the starting positions are increasing, and we give upper and lower bounds on the growth rate.

Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

Abstract Our main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k } $ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

Abstract The aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].

Determining which Fibonacci [formula omitted]-cubes can be [formula omitted]-transformation graphs

The Fibonacci (p, r)-cube is an interconnection topology, which includes a wide range of connection topologies as its special cases, such as the Fibonacci cube, the postal network, etc. Klavžar and Žigert [S. Klavžar, P. Žigert, Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are just the Z-transformation graphs (also called resonance graphs) of fibonaccenes, i.e. zigzag hexagonal chains. In this paper, we determine all Fibonacci (p, r)-cubes which can be the Z-transformation graphs of perfect matchings of plane (bipartite) graphs.

Tricyclic graphs with maximum Merrifield–Simmons index

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for the Merrifield–Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246–256]. The sharp upper bound for the Merrifield–Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield–Simmons index in ( n , n + 1 ) -graphs, J. Math. Chem. 43 (1) (2008) 75–91]. The sharp lower bound for the Merrifield–Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield–Simmons index is determined.

Generalizations of the Fibonacci and Lucas polynomials

In this note we consider two sequences of polynomials, which are denoted by {Un(k),m} and {Vn(k),m}, where k, m, n are nonnegative integers, and m ? 2. These sequences represent generalizations of the well-known Fibonacci and Lucas polynomials. For example, if m = 2, then we obtain exactly the Fibonacci and Lucas polynomials. If m = 3, then polynomials Un(k),3 and Vn (k),3 were considered in papers (G. B. Djordjevic, Fibonacci Quart. 39.2(2001), and G. B. Djordjevic, Fibonacci Quart. 43.4(2005)).

The Dying Rabbit Problem Revisited

AbstractIn this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in [Hoggat and Lind, Fibonacci Quart. 7: 482–487, 1969]) and we calculate explicitly their general term (extending the work in [Miles, Amer. Math. Monthly 67: 745–752, 1960]). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.

Fibonacci-like cubes as [formula omitted]-transformation graphs

The Fibonacci cube Γ n is a subgraph of n -dimensional hypercube induced by the vertices without two consecutive ones. Klavžar and Žigert [Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are precisely the Z -transformation graphs (or resonance graphs) of zigzag hexagonal chains. In this paper, we characterize plane bipartite graphs whose Z -transformation graphs are exactly Fibonacci cubes. If we delete from Γ n ( n ≥ 3 ) all the vertices with 1 both in the first and in the last position, we obtain the Lucas cube L n . We show, however, that none of the Lucas cubes are Z -transformation graphs, and characterize plane bipartite graphs whose Z -transformation graphs are L 2 k ′ for k ≥ 2 , which is obtained from L 2 k by adding two vertices and joining one to 1010 … 10 and the other to 0101 … 01 .

A generalization of Fibonacci and Lucas matrices

We define the matrix U n ( a , b , s ) of type s , whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix F n ( a , b , s ) , whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices ( s = 0 and s = 1 ). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix U n ( a , b , s ) is derived. In partial case we get the inverse of the generalized Fibonacci matrix F n ( a , b , 0 ) and later known results from [Gwang-Yeon Lee, Jin-Soo Kim, Sang-Gu Lee, Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices, Fibonacci Quart. 40 (2002) 203–211; P. Staˇnicaˇ, Cholesky factorizations of matrices associated with r -order recurrent sequences, Electron. J. Combin. Number Theory 5 (2) (2005) #A16] and [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)]. Correlations between the matrices U n ( a , b , s ) , F n ( a , b , s ) and the generalized Pascal matrices are considered. In the case a = 0 , b = 1 we get known result for Fibonacci matrices [Gwang-Yeon Lee, Jin-Soo Kim, Seong-Hoon Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527–534]. Analogous result for Lucas matrices, originated in [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)], can be derived in the partial case a = 2 , b = 1 . Some combinatorial identities involving generalized Fibonacci numbers are derived.

Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

Abstract It is well-known that Morgan-Voyce polynomials B n(x) and b n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $$ with $$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $$ with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > −¼B 2 in the third case. We show that the Diophantine equation with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

On the existence and on the number of [formula omitted]-kernels in the lexicographic product of graphs

In [G. Hopkins, W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, Fibonacci Quart. 22 (1984) 225–228.] and [I. Włoch, Generalized Fibonacci polynomial of graphs, Ars Combinatoria 68 (2003) 49–55] the total number of k -independent sets in the generalized lexicographic product of graphs was given. In this paper we study ( k , l ) -kernels (i.e. k -independent sets being l -dominating, simultaneously) in this product and we generalize some results from [A. Włoch, I. Włoch, The total number of maximal k -independent sets in the generalized lexicographic product of graphs, Ars Combinatoria 75 (2005) 163–170]. We give the necessary and sufficient conditions for the existence of ( k , l ) -kernels in it. Moreover, we construct formulas which calculate the number of all ( k , l ) -kernels, k -independent sets and l -dominating sets in the lexicographic product of graphs for all parameters k , l . The result concerning the total number of independent sets generalizes the Fibonacci polynomial of graphs. Also for special graphs we give some recurrence formulas.

A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers

In this paper, we construct the symmetric tridiagonal family of matrices M - α , - β ( k ) , k = 1 , 2 , … whose determinants form any linear subsequence of the Fibonacci numbers. Furthermore, we construct the symmetric tridiagonal family of matrices T - α , - β ( k ) , k = 1 , 2 , … whose determinants form any linear subsequence of the Lucas numbers. Thus we give a generalization of the presented in Cahill and Narayan (2004) [Cahill ND, Narayan DA. Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart 2004;42(3):216–21].

Fibonacci Numbers and Powers: 10765

Partial solution by the editors. We show that the conclusion holds if fn is not a kth power, where k > 2 and n > 4k 3, or if fn is a kth power and k is divisible by 2 or 3. (The assertion is false for k = 1.) First suppose that fn is a kth power for some k divisible by 2 or by 3. This makes fn a square or a cube. It is known that the only squares among the Fibonacci numbers are fi, f2, and f12 = 144 (J. H. E. Cohn, On square Fibonacci numbers, J. London Math. Soc. 39 (1964) 537-540), and that the only cubes among the Fibonacci numbers are fi, f2, and f6 = 8 (H. London and R. Finkelstein, On Fibonacci and Lucas numbers which are perfect powers, Fibonacci Quart. 7 (1969) 476-481, 487; errata, ibid. 8 (1970) 248). The assertion is easily checked in these cases. Now suppose that fn is not a perfect kth power, where k > 2 and n > 4k 3. Our plan is to show that !f/T-7 + kfn-2k is closer to VT than VT is to an integer. This is easy to check when n 12. We use the Binet formula for Fibonacci numbers: