We introduce a matricial Toeplitz transform and prove that the Toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. We investigate the injectivity of this transform and show how this distinguishes the Fibonacci sequence among other recurrence sequences. We then obtain new Fibonacci identities as an application of our transform.

In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, k-Jacobsthal and p-numbers, distance numbers, (2, k)-distance generalized (k, r)-Fibonacci in the sense by extending definition a recurrence relation with two adding three distance, simultaneously. Tiling combinatorial interp...

Recently we have been making studies [1, 2] on the Fibonacci number system that appears to be of considerable importance to the Fibonacci computer [3], A specific result that appears to be of particular interest is that through multilevel coefficients on a Fibonacci radix system, efficient extension of representations can occur. For example, through a ternary coefficient system, a doubling of t...

We use automata-theoretic approach to analyze properties of Fibonacci words. The directed acyclic subword graph (dawg) is a useful deterministic automaton accepting all suffixes of the word. We show that dawg’s of Fibonacci words have particularly simple structure. Our main result is a unifying framework for a large collection of relatively simple properties of Fibonacci words. The simple struc...

In [3], H. Belbachir and F. Bencherif generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. They prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations. [7], Mario Catalani define generalized bivariate polynomials, from which specifying initial conditi...

The Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamilt...

We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we ...

In this study we define and study the Bivariate Complex Fibonacci and Bivariate Complex Lucas Polynomials. We give generating function, Binet formula, explicit formula and partial derivation of these polynomials. By defining these bivariate polynomials for special cases Fn(x, 1) is the complex Fibonacci polynomials and Fn(1, 1) is the complex Fibonacci numbers. Finally in the last section we gi...

One says g is a Fibonacci primitive root modulo /?, wherep is a prime, iff g is a primitive root modulo/7 and g = g + 1 (mod p). In [1 ] , [2 ] , and [3] some interesting properties of Fibonacci primitive roots were developed. In this paper, we shall show that a necessary and sufficient condition for a prime/? ^ 5 to have a Fibonacci primitive root is p = 1 or 9 (mod 10) and Alp) = p 1, where/I...

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and search terms k-generalized Fibonacci sequence which repdigits. particular, find numbers powers 10 special case second part by using roots characteristic polynomial sequence, introduce tiny golden angles show feasibility new type in application to magnetic res...