scholarly journals Representation of Integers as Sums of Fibonacci and Lucas Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1625
Author(s):  
Ho Park ◽  
Bumkyu Cho ◽  
Durkbin Cho ◽  
Yung Duk Cho ◽  
Joonsang Park
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Elementary Problem ◽  
Representation Of Integers ◽  
Fibonacci And Lucas Numbers ◽  
Symmetric Properties

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.

Vieta-like products of nested radicals

2010 ◽  
Vol 94 (529) ◽  
pp. 62-66
Author(s):  
Thomas J. Osler
Keyword(s):  
Positive Integer ◽  
Recursion Relation ◽  
Golden Section ◽  
Infinite Product ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers ◽  
Image Position ◽  
Analytical Expressions

The beautiful infinite product of radicalsdue to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products:for N even, andfor N odd. Here N is a positive integer, FN and LN are the Fibonacci and Lucas numbers, and is the golden section. (The Fibonacci numbers are F1 = 1, F2 = 1, with the recursion relation , while the Lucas numbers are L1 = 1, L2 = 3 with the same recursion relation )

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

Trigonometry and Fibonacci numbers

2007 ◽  
Vol 91 (521) ◽  
pp. 216-226 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Fibonacci Numbers ◽  
Trigonometric Functions ◽  
Integer Sequences ◽  
Lucas Numbers ◽  
Mathematical Objects ◽  
Lucas Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Trigonometric Identities ◽  
The Way

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

scholarly journals A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jiangming Ma ◽  
Tao Qiu ◽  
Chengyuan He
Keyword(s):  
Fibonacci Numbers ◽  
Matrix Decomposition ◽  
Circulant Matrix ◽  
New Method ◽  
Circulant Matrices ◽  
Computational Time ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Jaroslav Seibert ◽  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Tridiagonal Matrix ◽  
Lucas Numbers ◽  
Tridiagonal Matrices ◽  
Fibonacci And Lucas Numbers

AbstractThe 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].

Graph-theoretic models for the Fibonacci family

2014 ◽  
Vol 98 (542) ◽  
pp. 256-265 ◽  
Author(s):  
Thomas Koshy
Keyword(s):  
Fibonacci Numbers ◽  
Mathematical Community ◽  
Large Integer ◽  
Lucas Numbers ◽  
Graph Theoretic ◽  
Fertile Ground ◽  
Graph Theoretic Models ◽  
Fibonacci And Lucas Numbers ◽  
Image Position

The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.Fibonacci numbersFnand Lucas numbersLnbelong to a large integer family {xn}, often defined by the recurrencexn=xn−1+xn−2, wherex1=a,x2=b, andn≥ 3. Whena=b= 1,xn=Fn; and whena= 1 andb= 3,xn=Ln. Clearly,F0= 0 andL0= 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:In this article, we will give a brief introduction to theQ-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.In 1960 C.H. King studied theQ-matrix

scholarly journals Some properties of the generalized Fibonacci and Lucas numbers

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2655-2665
Author(s):  
Gospava Djordjevic ◽  
Snezana Djordjevic
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Fibonacci And Lucas Numbers

In this paper we consider the generalized Fibonacci numbers Fn,m and the generalized Lucas numbers Ln,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

2021 ◽  
pp. 2150157
Author(s):  
Salim Badidja ◽  
Ahmed Ait Mokhtar ◽  
Özen Özer
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Representation Of Integers ◽  
Tribonacci Numbers ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
Positive Integers

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

scholarly journals Triple sums including Fibonacci numbers with three binomial coefficients

2021 ◽  
Vol 27 (1) ◽  
pp. 188-197
Author(s):  
Funda Taşdemir ◽  
Keyword(s):  
Linear Combination ◽  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

In this paper, we consider some triple sums that involve Fibonacci numbers with three binomial coefficients. We chose the indices of Fibonacci numbers as linear combination of the summation indices. Moreover, various types of alternating analogues of them whose powers depend on the index or indices are computed. These sums are evaluated in nice multiplication forms in terms of Fibonacci and Lucas numbers.

scholarly journals Fibonacci numbers in graphs with strong (1, 1, 2)-kernels

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Urszula Bednarz ◽  
Iwona Włoch
Keyword(s):  
Fibonacci Numbers ◽  
Complete Characterization ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper, we study strong (1, 1, 2)-kernels in graphs which generalize classical kernels, (1, 2)-kernels and 2-dominating kernels, simultaneously. We give a complete characterization of trees with a strong (1, 1, 2)-kernel. Moreover, we present some realization theorems and show connections between the number of strong (1, 1, 2)-kernels and Fibonacci and Lucas numbers.

Sign in / Sign up

Export Citation Format

Share Document