Fibonacci Numbers Research Articles

Golden lichtenberg algorithm: a fibonacci sequence approach applied to feature selection

Golden lichtenberg algorithm: a fibonacci sequence approach applied to feature selection

A new notion of convergence defined by weak Fibonacci lacunary statistical convergence in normed spaces

Abstract The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces.

Periods of Leonardo Sequences and Bivariate Gaussian Leonardo Polynomials

In this study, we investigate the periodic characteristics of Leonardo, Leonardo-Lucas, and Gaussian Leonardo sequences, presenting our findings through lemmas and theorems. Additionally, we introduce the concept of the power Leonardo like sequences and characterize the modules and integers within which these sequences exist. Furthermore, we conduct a comparative analysis between these power sequences and the power Fibonacci sequence under the same modulus. Lastly, we define a bivariate Gaussian Leonardo polynomial sequence and obtain specific properties associated with it.

Strong divisibility sequences and sieve methods

AbstractWe investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences.

The farfalle mystery

In 2017, for the \pi -day, Mickaël Launay described a remarkable property of the Fibonacci sequence, which he called “le mystère de la farfalle”. The name “farfalle” refers to a shape of pasta that is famous in France, a butterfly-shaped one, in connection with a likewise shape appearing in the numerical experiments. In these experiments, the Fibonacci numbers are put on the circle modulo N , and the segments joining consecutive elements of the sequence are drawn. Launay proposed a prize of 3.14 Euros to whom will explain the phenomenon he had discovered. We provide such an explanation, which illustrates recent works by Kurlberg and Rudnick and by Bourgain and Glibichuk.

On One Series of the Reciprocals of the Product of two Fibonacci Numbers Whose Indices Differ by an Even Number

This paper is inspired by a very interesting YouTube video by Michael Penn, professor of mathematics at Randolph College in Virginia, USA. He dedicated himself to the popularization of mathematics on his website in addition to his teaching and scientific work at the university and in addition to his scientific work. First, we deal with four specific series of the reciprocals of the product of two Fibonacci numbers whose indices differ by 2, 4, 6, and 8. Then, we generalize these four results to the series of the reciprocals of the product of two Fibonacci numbers whose indices differ by an even number. Finally, we perform a numerical verification of the derived formula using Maple 2020 software. Based on the derived formula, it can be concluded that the series we are dealing with belong to infinite series whose sum can be expressed in closed form.

Signal processing analysis for detection of anomalies in numerical series

It might be instinctively assumed that the occurrence of the first digit of a randomly selected number is uniformly distributed among 1 to 9. However, the Newcomb–Benford law (NBL), also known as the first-digit law or Benford’s law, reveals a completely different fact: the distributions follows a logarithm-type law. This bias in the frequency of occurrence of the leading digits of numbers is evident in everything from the lengths of rivers, the areas of countries, stock prices, and population numbers. Omnipresence of NBL emerges in mathematics (the Fibonacci sequence, the distribution of prime numbers, etc.) and also in the field of physics (i.e. the half-lives of unstable nuclei). From the earliest years, a large bibliography has been developed and applied to many fields such as fraud detection in finances, international trade and election results analysis. Usually, the diagnosis of such works is reduced to a qualitative binary assessment (conformity or non-conformity). The present paper aims to establish a supervised methodology to address existing gaps and enhance the reliability of final decisions. These gaps include uncertainty about the minimum data volume required to validate Benford’s Law, the low sensitivity of current metrics (Euclidean distance or Mean Absolute Difference) to detect minor levels of adulteration, and the need to quantify both the level and nature of adulteration. Two new procedures have been established coming from Signal Analysis and Processing theory. The first relates the deconvolution between the empirical leading digit distribution and the Benford one. The second procedure consists of obtaining the generating functions (via Hilbert and Fourier transforms) to increase the contrast between both functions: the empirical under research and Benford’s as the reference. Based on calibrated datasets, we are able to quantify the magnitude of accidental errors or malicious fraud adulteration and make the discrimination between both. Finally, as benchmark, fifteen real-world datasets were studied. This study shows that most of datasets are suspicious of Fraud. Notably, in the field of human health, several datasets demonstrate significant fraudulent deviations, defying conventional patterns of fraud and error. This phenomenon has been termed Berserker Fraud.

Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices

By considering the tiling of an N-board (a linear array of N square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers s n (where s n = ∑ i = 1 q v i s n − m i , s 0 = 1 , s n < 0 = 0 , where v i and m i are positive integers and m 1 < ⋯ < m q ) each raised to an arbitrary non-negative integer power. A ( w , g ; m ) -comb is a tile composed of m rectangular sub-tiles of dimensions w × 1 separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating s n 2 for the cases q = 2, v i = 1 , m 1 = M , m 2 = m + 1 for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are − 2 , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers.

Ascent sequences avoiding a triple of 3-letter patterns and Fibonacci numbers

Ascent sequences avoiding a triple of 3-letter patterns and Fibonacci numbers

FIBONACCI AND LUCAS NUMBERS AS PRODUCTS OF THEIR ARBITRARY TERMS

This study presents all solutions to the Diophantine equations F_k=L_m L_n and L_k=F_m F_n. To be clear, the Fibonacci numbers that are the product of two arbitrary Lucas numbers and the Lucas numbers that are the product of two arbitrary Fibonacci numbers are determined herein. The results under consideration are proven by using the Dujella-Pethő lemma in coordination with Matveev's theorem. All common terms of the Fibonacci and Lucas numbers are determined. Further, the Lucas-square Fibonacci and Fibonacci-square Lucas numbers are given.

On the F -Bernstein polynomials

UDC 517.5 We construct a new Bernstein operator, which is called the F -Bernstein operator obtained by using the F -factorial (Fibonacci factorial) and the Fibonomial (Fibonacci binomial). Then we examine the F -Bernstein basis polynomials and some of their properties. Moreover, we acquire certain connection between the F -Bernstein polynomials and the Fibonacci numbers.

Copper ratio obtained by generalizing the Fibonacci sequence

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.

A didactic engineering for the study of the Padovan’s combinatory model

Considering the content of history of mathematics textbooks, it’s evident that their emphasis is primarily on the illustrative aspects of recurring numerical sequences, with a particular focus on the Fibonacci sequence. Unfortunately, this limited approach results in the neglect of other sequences akin to the Fibonacci numbers, thus rendering the subject challenging for teaching purposes. This study aims to address this gap by offering a concise exploration of the combinatorial aspects of the Padovan numbers, specifically through the concept of a board as initially examined by mathematicians. In line with the research methodology of didactic engineering and the teaching theory of the theory of didactic situations, two problem situations have been developed, centered on the Padovan combinatorial model, thereby contributing to the enrichment of mathematical education within initial teacher training programs. Within this framework, various strategies are introduced that rely on visualization and counting, with the objective of illustrating specific mathematical identities suitable for potential classroom applications.

Narayana’s integer sequence revisited

An essential feature of the Fibonacci sequence (Fn): 1, 1, 2, 3, 5, 8, … is an explicit expression for its nth term Fn (Binet’s formula) and the corollary that is the golden ratio and {x} is the nearest integer to x.

The t − Fibonacci sequences in some finite groups with center Z ( G ) = G ′

We consider finitely presented groups \(G_{mn}\) as follows: \[ G_{mn}=\left\langle x, y \mid {x^{m}}={y^{n}}=1, {[x, y]^{x}}=[x, y], {[x, y]^{y}}=[x, y] \right\rangle ~~~~~m, n\ge 2. \] In this paper, we first study the groups \(G_{mn}\). Then by using the properties of \(G_{mm}\) and \(t-\)Fibonacci sequences in finitely generated groups, we show that the period of \(t-\)Fibonacci sequences in \(G_{mm}\) are a multiple of \(K(t, m)\). In particular for \(t \geq 3\) and \(p=2\), we prove \({{LEN}_{t}({{G}_{pp}})}= 2K(t,p)\).

A new graph labeling with Tribonacci, Fibonacci and Triangular numbers

In this paper, a new graph labeling technique using three sequences of numbers, namely Tribonacci, Fibonacci and Triangular is introduced. They are named as Tribo-Fibo-Triangular labeling and denoted as TFΔ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} labeling. Assignment of the labeling is done based on the families of generalized Petersen graphs GP (n,2), GP(n,3) and a generalization of these graph are also presented. Applications related to recruitment of employees in a health care sector and the selection of teams in a school ambience are arrived by using of these three sequences of graphs along with computational intelligence. A board game is developed and presented using these three sequences.

Benford's law and random integer decomposition with congruence stopping condition

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

Urine glucose concentration detection biosensor using one-dimensional photonic crystals with periodical and Fibonacci sequences based on Tamm plasmon resonance

This paper presents the design of biosensors utilizing one-dimensional photonic crystals with periodical and Fibonacci sequences for measuring glucose concentration in urine, aimed at facilitating continuous blood glucose monitoring for diabetic patients. Exploiting Tamm plasmon resonance within a photonic band gap in the medium wave infrared band, the biosensor comprises a configuration with a one-dimensional photonic crystal and an Ag layer deposited on an infrared prism, with a urine sample layer in between. Utilizing the transfer matrix method, the reflection spectra for electromagnetic waves are calculated. The wavelength position of the Tamm plasmon resonant dip is influenced by variations in glucose concentration within the urine sample. This is attributed to the distinct refractive indices exhibited by urine samples with different glucose concentrations. Optimizing biosensor performance under various incident angles involves adjusting the Ag layer and urine sample thicknesses while maintaining excellent linear characteristics. The optimal performance of the biosensor with Fibonacci sequence one-dimensional photonic crystal is significantly superior, with a sensitivity of 113,000 nm RIU−1, a figure of merit of 2.05 × 105 RIU−1, and a detection limit of 4.84 × 10−7 RIU. The combination of high performance and a straightforward structure makes the proposed biosensors for detecting urine glucose concentrations promising in biomedical diagnostics.

Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers

Abstract Let (Fn ) n≥0 and (Ln ) n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10 d Fm + Lk and Fn = 10 d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk , respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.

Some general summation results with applications to arctangent-type series

This article extends the telescopic summation and Abel summation by parts methods by introducing various tuning parameters and examining sequences of variable-dependent functions. In a substantial part, these general results are applied to arctangent-type series. We re-examine known results from a fresh viewpoint and establish numerous new ones, with an emphasis on establishing unified formulas applicable in various scenarios. In particular, some new arctangent-type series defined with the binomial coefficients and others with the Fibonacci sequence are demonstrated. Therefore, this work contributes to the advancement of telescopic techniques and enriches our understanding of arctangent-type series.