The focus of this paper is to study the HOMFLY polynomial of (2, n)-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of (2, n)-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental ...

The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent sets of the h power of a path ordered by inclusion. For h = 1 such a diagram is called a Fibonacci cube, and for h > 1 we obtain a generalization of th...

We study the propagation of electromagnetic waves in one-dimensional quasiperiodic systems and its dispersion relation for plane waves and for waveguide structures. In the photonic band gaps, periodic, Fibonacci, and Thue-Morse multilayer systems can be described by a complex effective wave vector. Its negative imaginary part causes an exponential decay of the transmission coefficient due to a ...

This paper presents a comparative analysis of several matrix analogs of the Diffie-Hellman algorithm, namely, Yerosh-Skuratov and Megrelishvili protocols, as well as alternative protocols based on irreducible polynomials and primitive Galois or Fibonacci matrices. Binary matrix is primitive, if the sequence of its powers in the ring of residues mod 2 forms a sequence of maximum length ( m  seq...

We prove that the Fibonacci word f satisfies among all characteristic Sturmian words, three interesting extremal properties. The first concerns the length and the second the minimal period of its palindromic prefixes. Each of these two properties characterizes f up to a renaming of its letters. A third property concerns the number of occurrences of the letter b in its palindromic prefixes. It c...

A closed-form expression is derived for the enumeration of all palindromic binary strings of length n > r having no r-runs of 1’s, in terms of the r-Fibonacci sequence. A similar closed-form expression for the number of zeros contained in all such palindromic binary strings is derived in terms of the number of zeros contained in all binary strings having no r-runs of 1’s.

In this article we study a random walk on a particularly simple graph. This walk is determined by a probabilistic process associated with the Fibonacci sequence. Exact formulas are derived for the expected proportions of time spent on each arc of the graph for a walk of length n, giving rise to sequences that do not appear in Sloane’s On-Line Encyclopedia of Integer Sequences. We also obtain as...

Inspired by the definition of the barred pattern-avoiding permutation, we introduce the new concept of dotted pattern for permutations. We investigate permutations classes avoiding dotted patterns of length at most 3, possibly along with other classical patterns. We deduce some enumerating results which allow us to exhibit new families of permutations counted by the classical sequences: 2n, Cat...

Quantum cryptography is commonly used to generate fresh secure keys with quantum signal transmission for instant use between two parties. However, research shows that the relatively low key generation rate hinders its practical use where a symmetric cryptography component consumes the shared key. That is, the security of the symmetric cryptography demands frequent rate of key updates, which lea...

Trapezoidal words are words having at most n+1 distinct factors of length n for every n ≥ 0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a p...