Sequences and Polynomial Congruence

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

It is shown that period, rank and order of the ) , ( b a -Fibonacci sequence remain invariant by extension to the generalized ) , , , ( d c b a -Lucas sequence provided a specific determinant is relatively prime to the modulus.Given an odd prime modulus p and the 1  p exceptional ) , ( b a -Lucas sequences, for which p divides the discriminant of these sequences, we determine the corresponding periods and relate them to those of the ) , ( b a -Fibonacci sequence.The results can be viewed as extended versions of classical results by Wall in case ) 1 , 1 ( ) , (  b a .

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

Efficient quantum key distribution using Fibonacci-number coding with a biased basis choice

It is well known that the companion sequence of the Fibonacci sequence is Lucas's sequence. For the generalized Fibonacci sequences, the companion sequence is not unique. Several authors proposed different definitions, and they are in a certain sense all good. Our purpose is to introduce a family of companion sequences for some generalized Fibonacci sequence: the $r$-Fibonacci sequence. We evaluate the generating functions and give some applications, and we exhibit convolution relations that generalize some known identities such as Cassini's. Afterwards, we calculate the sums of their terms using matrix methods. Next, we propose a $q$-analogue and extend the definition to negative $n$s. Also, we define the incomplete associated sequences using a Euler--Seidel-like approach.

The Horadam sequence Hn(a,b;p,q) unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials (v1,v2,v3) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices kn through a parameterized grammar Gk. Using the modified grammar Gk˜, we derive convolution formulas involving the weighting term (−q)n−i. Furthermore, applying the proposed framework to (p,q)-Fibonacci and (p,q)-Lucas sequences, we derive explicit convolution formulas with parameters (p,q). The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers.

In this paper we illustrate one of the most fascin ating things about the Fibonacci numbers: their connection to nature. HISTORICAL PERSPECTIVE The greatest European mathematician of the middle ages, his full name was Leonardo of Pisa, now known as Fibonacci short for filius Bonacci . Fibonacci's mathematical contribution is memorial. He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a sy mbol for zero: 1 2 3 4 5 6 7 8 9 0 In 1202, his book on to do arithmetic in the de cimal system is called Liber abbaci (meaning Book of the Abacus or Book of Calculating ). Fibonacci and the Fibonacci sequence are prime examples of how mathematics is connected to seemingly unrelated thi ngs. Each term in the Fibonacci sequence is called a Fibonacci number. Even though these were introduced in 1202, they remain fascinating and mysterious to people to day. Fibonacci gave a problem in his book discovering th e magical by analyzing the birth rate of rabbits. Fibonacci's Problem: Someone placed a pair of rabbits in a certain plac e, enclosed on all sides by a wall, to find out many pairs will be born in the course of on e year, it being assumed that every month a pair of rabbits produces another pair, and that r abbits begin to bear young two months after their own birth. Fibonacci's Solution : The Fibonacci Sequence! Even if the growth rabbit's population problem is f ar from being a biologically realistic problem, rabbits helped Fibonacci to discover one o f the great marvels of nature. MATHEMATICAL AND COMPUTATIONAL APPROACH The most ubiquitous, and perhaps the most intriguin g, number pattern in mathematics is Fibonacci sequence. The pattern of the numbers of life is elegantly simple. The first number of the sequence is 0, the second number is 1, and e ach subsequent number is equal to the sum of the previous two of the sequence itself. In mathematical terms, it is defined by the following recurrence relation:

This dissertation is a study on Generating Functions and the Linear Recurrence Relations and is divided in three chapters in the following manner:On the first chapter we present Ordinary Generating Functions and Exponential Generating Functions along with their definitions, properties and a few theorems with their respective demonstrations.Still on this chapter we explore the concepts of Partitions and some Partition Generating Functions.On the second chapter we explore the concept of Recurrence Relations, specifically the Linear ones.We show a few methods for the resolution of Recurrence Relations, highlighting the resolution based on Generating Functions and also an interesting method using matrix diagonalization.The methods are very well detailed and are easy to comprehend.Chapter 3 is made of four applications on Recurrence Relations and Generating Functions.(n,k)-block fountain is a method of piling up coins on a plane obeying a few criteria.Steiner's pizza consists of dividing the pizza and obtaining the greatest number of slices with n cuts, triangulation of a convex n-agon is a problem of diving polygons in triangles and lastly one of the most famous sequences in Mathematics, the Fibonacci Sequence, where we show the sequence recurrence relation and its respective generating function.

On the spacings between C-nomial coefficients

In this paper, we give explicit formulas for elements of the Fibonacci, and Lucas Pascal triangles. The structure of these objects and Pascal's original triangle coincide. Keeping the rule of addition, we replace both legs of the Pascal triangle by the Fibonacci sequence, and the Lucas sequence, respectively. At the end of the study we describe how to determine such a formula for any binary recurrence $\{G_n\}^{\infty}_{n=0}$ satisfying $G_n=G_{n−1}+G_{n−2}$. Other scattered results are also presented.

Mathematical iteration is a process for generating a sequence in which one or more initial terms are given and each subsequent term is determined from its predecessors in the same way. An equation that describes the relationship between a term and its predecessors is called a recurrence relation. Arithmetic and geometric sequences, common topics in high school algebra courses, are examples of iterative processes. Arithmetic sequences are generated iteratively from an initial term, a1, a common difference, d, and a recurrence relation, an+1, = an+ d. Geometric sequences are generated from an initial term, a1 a common ratio, r, and a recurrence relation, an+1 =an •r. Mathematical iteration is used in many other mathematical situations and algorithms, such as the Fibonacci sequence, the Euclidean algorithm, and Newton's method for solving equations.

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for Fp, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

The Binet forms for the well known Fibonacci and Lucas Sequences {Fn} and {Ln}are discussed, in detail, in this paper.

The Braxton-Shore house has been considered as an exemplar of 'space architecture' – an architecture of 'space forms' which R. M. Schindler believed distinguished his work from contemporary functionalists and internationalists. To achieve such' space architecture', Schindler utilized a proportional system of a space reference frame as a mental tool. This paper first outlines the spatial scheme of the Braxton-Shore house, explains Schindler's space reference frame, and finally interprets the proportional design of the house in analogy with the 'row' system, the Fibonacci and Lucas sequence, and musical ratios.