Generalized Gaussian Fibonacci Numbers and its Determinantal Identities

In this paper, we present the determinantal identities of generalized Gaussian Fibonacci numbers. The generalized Gaussian Fibonacci sequence is defined by the recurrence relation. This was introduced by S. Pethe and A. F. Horadam. Also, we present its determinantal identities with classical numbers like gaussian Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, jacobsthal-Lucas, Bronze, Nickel and Mersenne numbers.

Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.

A symmetric algorithm for hyperharmonic and Fibonacci numbers

In the present paper, we first study the Gaussian Leonardo numbers and Gaussian Leonardo hybrid numbers. We give some new results for the Gaussian Leonardo numbers, including relations with the Gaussian Fibonacci and Gaussian Lucas numbers, and also give some new results for the Gaussian Leonardo hybrid numbers, including relations with the Gaussian Fibonacci and Gaussian Lucas hybrid numbers. For the proofs, we use the symmetric and antisymmetric properties of the Fibonacci and Lucas numbers. Then, we introduce the Gaussian Leonardo polynomials, which can be considered as a generalization of the Gaussian Leonardo numbers. After that, we introduce the Gaussian Leonardo hybrid polynomials, using the Gaussian Leonardo polynomials as coefficients instead of real numbers in hybrid numbers. Moreover, we obtain the recurrence relations, generating functions, Binet-like formulas, Vajda-like identities, Catalan-like identities, Cassini-like identities, and d’Ocagne-like identities for the Gaussian Leonardo polynomials and hybrid polynomials, respectively.

Lucas and Gibonacci numbers are two sequences of numbers derived from a welknown numbers, Fibonacci numbers. The difference between Lucas and Fibonacci numbers only lies on the first and second elements. The first element in Lucas numbers is 2 and the second is 1, and nth element, n ≥ 3 determined by similar pattern as in the Fibonacci numbers, i.e : Ln = Ln-1 + Ln-2. Gibonacci numbers G0 , G1 ,G2 , ...; Gn = Gn-1 + Gn-2 are generalized of Fibonacci numbers, and those numbers are nonnegative integers. If G0 = 1 and G1 = 1, then the numbers are the wellknown Fibonacci numbers, and if G0 = 2 and G1 = 1, the numbers are Lucas numbers. Thus, the difference of those three sequences of numbers only lies on the first and second of the elements in the sequences. For Fibonacci numbers there are quite a lot identities already explored, including the sum of cubes, but there have no discussions yet about the sum of cubes for Lucas and Gibonacci numbers. In this study the sum of cubes of Lucas and Gibonacci numbers will be discussed and showed that the sum of cubes for Lucas numbers is and for Gibonacci numbers is

Fibonaccene.- On a Class of Numbers Related to Both the Fibonacci and Pell Numbers.- A Property of Unit Digits of Fibonacci Numbers.- Some Properties of the Distributions of Order k.- Convolutions for Pell Polynomials.- Cyclotomy-Generated Polynomials of Fibonacci Type.- On Generalized Fibonacci Process.- Fibonacci Numbers of Graphs III: Planted Plane Trees.- A Distribution Property of Second-Order Linear Recurrences.- On Lucas Pseudoprimes which are Products of s Primes.- Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Networks and in Electric Line Theory.- Infinite Series Summation Involving Reciprocals of Pell Polynomials.- Fibonacci and Lucas Numbers and Aitken Acceleration.- On Sequences having Third-Order Recurrence Relations.- On the Solution of the Equation G n = P(x).- Distributions and Fibonacci Polynomials of Order k, Longest Runs, and Reliability of Consecutive-k-Out-Of-n : F Systems.- Fibonacci-Type Polynomials and Pascal Triangles of Order k.- A Note on Fibonacci and Related Numbers in the Theory of 2 x 2 Matrices.- Problems on Fibonacci Numbers and Their Generalizations.- Linear Recurrences having almost all Primes as Maximal Divisors.- On the Asymptotic Distribution of Linear Recurrence Sequences.- Golden Hops Around a Circle.

Nash [4] used recursive sequences like the Fibonacci numbers, {Fn}, to investigate factors and divisibility. The Fibonacci numbers are defined by $$ \begin{array}{*{20}{c}} {{F_n} = {F_{n - 1}} + {F_{n - 2}},}&{n > 2,} \end{array} $$ (1.1) with F1 = F2 = 1. (The Lucas numbers, L n , satisfy the same linear homogeneous recurrence relation (1.1) but have initial conditions L 1, =1, L 2 = 3.) Brillhart, Montgomery and Silverman [1] also used the Fibonacci and Lucas numbers and the identity $$ {F_{2n}} = {F_n}{L_n} $$ (1.2) to investigate factorizations. Shannon, Loh and Horadam [8] generalized this in the context of the functional equation $$ f(2k - {x^2}) = f(x)f( - x) $$ (1.3) when k = 1. One of the authors (RPL) has attempted to find irreducible polynomial solutions over Q of degree n to the relation (1.3) and found that for k = 0, 1, nearly all solutions are proper divisors of recurrence relations. It is the purpose of this paper to draw some of the strands of this study together. In the next two sections we look at sequences which satisfy (1.3) when k = 0 and k = 1. Then we take some computer generated examples to consider aspects of the functions for general k.

In this paper we prove an identity between sums of reciprocals of Fibonacci and Lucas numbers. The Fibonacci numbers are defined for all n ≥ 0 by the recurrence relation Fn + 1 = Fn + Fn-1 for n ≥ 1, where F0 = 0 and F1 = 0. The Lucas numbers Ln are defined for all n ≥ 0 by the same recurrence relation, where L0 = 2 and L1 = 1 We prove the following identify.

Fibonacci and Lucas numbers are special number sequences that have been the subject of many studies throughout history due to the relations they provide. The studies are continuing today, and findings about these number sequences are constantly increasing. The relations between the Fibonacci and Lucas numbers, which were found during the proof of the prime between two consecutive numbers belonging to the Fibonacci or Lucas number sequence with the Euclidean algorithm, started our project. In the project, Diophantine equations whose coefficients are Lucas or Fibonacci numbers have been studied, various relations have been found, and their proofs have been made. \begin{align*} F_nx-F_{n+1}y & =(-1)^n, \\ L_nx-L_{n-1}y & =1. \end{align*} As in the above example, the equivalents of $x$ and $y$ values were found in the Diophantine equations with Fibonacci and Lucas number coefficients; and based on this example, different variations of the Diophantine equations whose coefficients were selected from the Fibonacci and Lucas number sequences were created, and their proofs were made. Secondly, the geometric shapes consisting of vertices determined by pair of numbers selected from the Fibonacci or Lucas number sequence were considered, and their properties were examined. Various relations were found between them, and generalizations were made.

S equences have been fascinating topic for mathematicians for centuries. The Fibonacci sequences are a source of many nice and interesting identities . A similar interpretation exists for Lucas sequence. The Fibonacci number, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula and , where are an n th number of sequences. The Lucas Sequence is defined by the recurrence formula and , where an nth number of sequences are. In this paper, we present generalized Fibonacci-Lucas sequence that is defined by the recurrence relation , with B0= 2s, B1 = s . We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binets formula and other simple methods. Keywords : Fibonacci sequence, Lucas Sequence, Generalized Fibonacci sequence, Binets Formula.

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula and F0=0, F1=1, where Fn is a nth number of sequence. The Lucas Sequence is defined by the recurrence formula and L0=2, L1=1, where Ln is a nth number of sequence. In this paper, Generalized Fibonacci-Lucas sequence is introduced and defined by the recurrence relation with B0 = 2b, B1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet’s formula and other simple methods.

Basic properties of the golden ratio geometric problems in two dimensions geometric problems in three dimensions Fibonacci numbers Lucas numbers and generalized Fibonacci numbers continued fractions and rational approximants generalized Fibonacci representation theorems optimal spacing and search algorithms commensurate and incommensurate projections Penrose tilings quasicrystallography biological applications construction of the regular pentagon the first 100 Fibonacci and Lucas numbers relationships involving Fibonacci and Lucas numbers.

The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, , and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Bn=Bn-1+Bn-2, with B0=2s, B1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet’s formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicomplex Gaussian Fibonacci, the bicomplex Gaussian Lucas, Gaussian Fibonacci, Gaussian Lucas and Fibonacci numbers.

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers