Fibonacci Numbers Research Articles

MODELO SEMÂNTICO DE OPERAÇÕES ARITMÉTICAS E LÓGICAS PARA HARDWARE VIRTUAL

The objective of this project was to develop a semantic model of arithmetic and logical operations for virtual hardware; this goal was achieved through three main steps: Implement logic gates and arithmetic and logic units in Nand2tetris virtual hardware simulator software; Implement recursive and iterative functions to check the correct functioning of logical and arithmetic operations in the p-code machine virtual machine. In the first stage, implementations of the elementary logic gates were developed in the hardware simulator software Nand2tetris. From these logic gates, combinational circuits and sequential circuits were built as logical and arithmetic unit, Half Adder, Full Adder, 16-bit Addition, 16-bit logical negation, 16-bit logical AND, 16-bit logical OR, among others. The second stage involved experimenting at a high level with arithmetic and logical operations, provided by the virtual machine p-code machine, which was implemented in the C programming language, as addition, subtraction, greater than, less than. In the third stage, having already implemented logical and arithmetic operations, iterative and recursive programs were developed to calculate the nth value of the Fibonacci Sequence and the factorial of any number n.

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained.

An upper bound on asymptotic repetition threshold of balanced sequences via colouring of the Fibonacci sequence

We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetition threshold of d-ary balanced sequences. The bound is attained for d=2,4 and 8 and we conjecture that it happens for infinitely many even d's.Our bound reveals an essential difference in behaviour of the repetition threshold and the asymptotic repetition threshold of balanced sequences. The repetition threshold of d-ary balanced sequences is known to be at least 1+1d−2 for each d≥3. In contrast, our bound implies that the asymptotic repetition threshold of d-ary balanced sequences is at most 1+τ32d−3 for each d≥2, where τ is the golden mean.

On 3-parameter quaternions with higher order generalized Fibonacci numbers components

On 3-parameter quaternions with higher order generalized Fibonacci numbers components

Fiboquadratic numbers and Rithmomachia

Abstract This paper shows our discovery of the fiboquadratic numbers, quadratic Fibonacci numbers in rithmomachia. This board game consists in performing mathematical operations to attack an opponent through geometric numbered pieces. We will show that the numbers on the board pieces are the first terms of six sequences that we will define through a recursive pattern of superpartients. We then prove that this extension is the solution of a recurrence relation with initial conditions, and therefore, is unique. The fiboquadratic numbers we introduce in this work are not limited to the context of this board. In general, we can generate any fiboquadratic sequence with arbitrary initial conditions.

A General Convolution Identity

Summary The convolution of the Fibonacci numbers and the Lucas numbers is equal to a simple multiple of a Fibonacci number. We show how this can be generalized to other Fibonacci-type and Lucas-type numbers. Along the way, we discuss generating functions and Newton’s Identities.

Mathematical Approaches to the Study of Human Information Processing

Introduction. We consider the Golden Ratio principle and the Fibonacci sequence in terms of its possible use in the production and perception of information. The correspondence between the Fibonacci sequence and the principle of symmetry is most clearly shown when numbers are replaced with concepts and judgments.Methodology and sources. Language is an integral part not only of consciousness, but also of reality, and therefore it is quite natural to assume that its formation and development took place in accordance with the objective laws of nature that ensure the effectiveness of the interaction of structural and systemic formations as part of the whole. Such an approach makes it possible to find answers to questions that have not been solved within the narrow linguistic framework of the study, and also contributes to the fact that language structures designed to describe the state of affairs in reality provide a methodological basis for cognition at the disposal of a person.Results and discussion. The simplest language phrases built on the models of «what is what», «what is which», «what/who does what», already contain all the essential elements of thought: 1) the separation of two objects; 2) comparing them with each other (in consciousness), and 3) inference (stating their separateness and differences from each other). Thus, symmetry and asymmetry are the essence of the process of obtaining new knowledge in any field of research. Each new judgment in the chain constructed in accordance with the Fibonacci sequence includes the two previous ones, but in a new quality – based on their synthesis. At the same time, all copies of the original and received structures remain in memory.Conclusion. The search for laws implies not only generalization of ideas, but also finding the simplest schemes according to which material objects are created and processes occurring in the natural environment are implemented. In accordance with the processes regulating the stability and variability of systems in the natural environment, the activity of consciousness is also carried out, aimed at obtaining knowledge and their formation on the basis of symmetry and asymmetry. Symmetry contributes to the preservation and assimilation of the original data, asymmetry allows you to extract new information based on them. The information coming from outside is transformed into linguistic structures – the only way possible for a person to reflect the results of logical and sensory analysis.

Patterns among square roots of the 2 × 2 identity matrix

The 2 × 2 identity matrix, $${I_2} = \left( \begin{gathered}{\rm{1 \,\,\,0}} \hfill \\ {\rm{0 \,\,\,1}} \hfill \\ \end{gathered} \right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.

From the Fibonacci Icosagrid to E8 (Part II): The Composite Mapping of the Cores

This paper is part of a series that describes the Fibonacci icosagrid quasicrystal (FIG) and its relation to the E8 root lattice. The FIG was originally constructed to represent the intersection points of an icosahedrally symmetric collection of planar grids in three dimensions, with the grid spacing of each following a Fibonacci chain. It was found to be closely related to a five-fold compound of 3D sections taken from the 4D Elser–Sloane quasicrystal (ESQC), which is derived via a cut-and-project process from E8. More recently, a direct cut-and-project from E8 has been found which yields the FIG (presented in another paper of this series). The present paper focuses not on the full quasicrystal, but on the relationship between the root polytope of E8 (Gosset’s 421 polytope) and the core polyhedron generated in the FIG, a compound of 20 tetrahedra referred to simply as a 20-Group. In particular, the H3 symmetry of the FIG can be seen as a five-fold or “golden” composition of tetrahedral symmetry (referring to the characteristic appearance of the golden ratio). This is shown to mirror a connection between tetrahedral and five-fold symmetries present in the 421. Indeed, the rotations that connect tetrahedra contained within the 421 are shown to induce, in a certain natural way, the tetrahedron orientations in the 20-Group.

Development and evolution of the Asteraceae capitulum.

Asteraceae represent one of the largest and most diverse families of plants. The evolutionary success of this family has largely been contributed to their unique inflorescences, capitula that mimic solitary flowers but are typically aggregates of multiple florets. Here, we summarize the recent molecular and genetic level studies that have promoted our understanding of the development and evolution of capitula. We focus on new results on patterning of the enlarged meristem resulting in the iconic phyllotactic arrangement of florets in Fibonacci numbers of spirals. We also summarize the current understanding of the genetic networks regulating the characteristic reproductive traits in the family such as floral dimorphism and differentiation of highly specialized floral organs. So far, developmental studies in Asteraceae are still limited to a very narrow selection of model species. Along with the recent advancements in genomics and phylogenomics, Asteraceae and its relatives provide an outstanding model clade for extended evo-devo studies to exploit the morphological diversity and the underlying molecular networks and to translate this knowledge to the breeding of the key crops in the family.

PhiΦBreast New Diagnostic Techniques for Breast Cancer Detection

Background: Today, breast cancer is one of the most aggressive cancers in women and new cases continue to increase worldwide. In order to reduce mortality, we need to detect this life-threatening disease at an earlier stage. For two years, we have conducted a study for the identification and characterization of suspicious breast lesions using a new diagnostic technique applied to ultrasonography and mammography called “PhiΦBreast.” Methods: We have conducted a study for the identification and characterization of suspicious breast lesions using a new diagnostic technique applied to mammography and ultrasonography. Ultrasound systems with high frequency probes was used, 10-15 MHz, and Digital Breast Tomosynthesis (DBT). Identification and characterization of category C4-C5 lesions of the breast with high Predictive Positive PPV value, with a new innovative method called “PhiΦBreast” using the Golden Ratio (Phi, or Φ 1.618...) Fibonacci sequence and a Predictive Algorithm, applied to the ultrasonography and mammography with subsequent deepening with cytological examination using Fine Needle Aspiration Cytology (FNAC), according to evaluation criteria of the Breast Imaging Report Data System (BI-RADS) and the American College of Radiology (ACR). Usefulness of this research and the use of this new diagnostic tecnique is to detect the breast cancer in early stage. In addition to develop a classification model of the histological type identified in the section areas and the percentage of probability in relation between the Golden Ratio (Φ) and Fibonacci sequence. Results: With the use of Golden Ratio and Fibonacci sequence, applied to ultrasonography and mammography, we have experimented and developed a diagnostic map with characteristics of high probability of identifying suspicious lesions at an early stage. We examined 987 women, 55 lesions detected with PhiΦBreast pattern were classified according to BI-RADS descriptors for US-imaging, including morphologic features that had a high predictive value for the malignancy (p <0.001) This innovative diagnostic technique has shown a sensitivity of 95%, a specificity of 97%, a positive predictive value (PPV) of 97%, and negative predictive value (NPV) of 96%. Furthermore, with a Predictive Algorithm associated with malignant cytology after FNAC, we have classified different types of potentially life-threatening cancers for patients. Conclusion: PhiΦBreast could be an important new model diagnostic technique to be applied mammography and ultrasound for detection of malignant lesions of category C4-C5. In diagnostic imaging is fundamental to recognize predictively the characteristics of a potentially aggressive tumor. This new model could represent the cornerstone as an important contribution for early diagnosis of breast cancer.

Arctangent Series on Pell and Pell-Lucas Polynomials via Cassini-Like Formulae

By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.

Fibonacci sequences and the aesthetics of nature’s forms in modern design

Abstract This study conducts a comprehensive examination of the Fibonacci sequence’s mathematical model and its integration into contemporary design frameworks. It delineates the morphological aesthetic principles inherent in modern design, accentuating the visual aesthetic attributes governed by the Fibonacci sequence. The investigation further delves into color and contrast as principal aesthetic elements influenced by this sequence. The paper provides a brief analysis of the Fibonacci sequence’s impact on artistic and design practices across Eastern and Western cultures, introducing a concept of art and design communication. A least-squares measurement model is constructed to assess the impact of digital development and communicative elements on public receptivity towards design works that incorporate the Fibonacci sequence. Selected modern sofa designs based on the Fibonacci sequence serve as the focal point of this research, examining market reception and consumer preferences for the embedded mathematical aesthetics. The findings reveal a marked preference for monochromatic sofa upholstery in both Eastern and Western regions, underscoring a significant consistency. This preference suggests that the integration of Fibonacci-based mathematical aesthetics into modern product design is not only viable but also enhances the visual appeal and color coordination of the products.

More generalized k(ε)-Fibonacci sequence, series, and its applications

In this study, we present a generalized higher-order delta operator with the co-efficient of falling factorial and its inverse, both of which allow us to get more generalized k(ε)-Fibonacci sequences along with their sums, a few theorems, and some intriguing conclusions regarding the sum of more generalized terms of k(ε)-Fibonacci sequences with falling factorial. In addition, the n-fold Fibonacci ratio and a few applications in life sciences and crystal growth were presented in the work. In addition, suitable examples are presented using MATLAB to show our results.

An efficient confidentiality scheme based on quadratic chaotic map and Fibonacci sequence

<p>In today's rapidly evolving digital landscape, secure data transmission and exchange are crucial for protecting sensitive information across personal, financial, and global infrastructures. Traditional cryptographic algorithms like RSA and AES face increasing challenges due to the rise of quantum computing and enhanced computational power, necessitating innovative approaches for data security. We explored a novel encryption scheme leveraging the quadratic chaotic map (QCM) integrated with the Fibonacci sequence, addressing key sensitivity, periodicity, and computational efficiency. By employing chaotic systems' inherent unpredictability and sensitivity to initial conditions, the proposed method generates highly secure and unpredictable ciphers suitable for text and image encryption. We incorporated a combined sequence from the Fibonacci sequence and QCM, providing enhanced complexity and security. Comprehensive experimental analyses, including noise and occlusion attack simulations, demonstrate the scheme's robustness, resilience, and practicality. The results indicated that the proposed encryption framework offers a secure, efficient, and adaptable solution for digital data protection against modern computational threats.</p>

An Analysis of the Application of Motion Capture Technology in the Creation of Contemporary Dance with the Integration of Fibonacci Sequences

Abstract This paper combines the existing development of Fibonacci sequence aesthetics in art creation for the aesthetic planning of the Fibonacci sequence on music rhythm, proposes to design a new modern dance creation method by combining the diversified rotation methods of the Fibonacci sequence, and builds a process framework for generating Fibonacci sequence dance score based on motion capture. The wavelet transform is employed to determine the start of the dance movement, and the gradient descent algorithm is utilized to analyze the time coordinates and refine the dance movement. Recognize the segmented elemental action sequences using the DTW method, propose three basic actions from the dataset, and test the recognition rate of the DTW algorithm for each action. Generate a Fibonacci sequence dance score by combining human gestures. Combined with the research scoring, analyze the artistic talent of the Fibonacci sequence-generated dance. According to the data results, it can be seen that the action feature vector of the two-handed waving image is P=[0.8968, 0.9925, 0.9124, 1.0243, 1.0198, 1.2438, 1.4596, 1.3698, 1.5868, 1.6596, 1.5029] with 98.67% of the two-handed waving recognition rate (with a standard deviation of 0.25). The evaluation of Fibonacci sequence generation dance by research users tends to be moderate.

Sequences from Fibonacci to Catalan: A combinatorial interpretationviaDyck paths

We use Dyck paths having some restrictions in order to give a combinatorial interpretation for some famous number sequences. Starting from the Fibonacci numbers we show how thek-generalized Fibonacci numbers, the powers of 2, the Pell numbers, thek-generalized Pell numbers and the even-indexed Fibonacci numbers can be obtained by means of constraints on the number of consecutive valleys (at a given height) of the Dyck paths. By acting on the maximum height of the paths we get a succession of number sequences whose limit is the sequence of Catalan numbers. For these numbers we obtain a family of interesting relations including afull historyrecurrence relation. The whole study can be accomplished also by involving particular sets of stringsviaa simple encoding of Dyck paths.

Distribution Generated by a Random Inhomogenous Fibonacci Sequence

Let G0=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_0=0$$\\end{document} and G1=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_1=1$$\\end{document}. The present study deals with the inhomogeneous version Gn=Gn-1+Gn-2+wn-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} G_n=G_{n-1}+G_{n-2}+w_{n-2} \\end{aligned}$$\\end{document}of the Fibonacci sequence, where wn-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w_{n-2}$$\\end{document} takes value a with probability p, and does value b with 1-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1-p$$\\end{document}. We describe the probability distribution of the values of Gn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_n$$\\end{document} with fixed n, and examine the properties like expected value and variance. The most challenging feature is the fractal-like structure of the distribution.

An Effective Method for Compute the Roughness of Fractal Facades Based on Box-Counting Dimension (Db)

Benoit Mandelbrot coined the word “fractal” in the late 1970s, but an object is now defined as fractals in form known to artists and mathematicians for centuries. A fractal object is self-similar in that the subsections of the object are somewhat similar to the whole object. No matter how small the subdivision is, the subsection contains no less detail than the whole. Atypical example of a fractal body is the “snowflake curve” (invented by Helga von Koch (1870-1924) in 1904. There are as many relationships between architecture, the arts, and mathematics as symmetry. The golden ratio, the Fibonacci sequence in this paper explain the method of counting box and measuring the roughness ratio. And small scale analysis after calculating the box to understand fractal concepts, we must know two dimensions. Through analyzing the samples in the research, it has been proven that fractal geometry is present everywhere in our lives in nature, in buildings, and even in plants and its role in architecture is to find fractal systems that appeal to our inclinations for dynamic vitality. Therefore, finding such fractals enables us to create high-performance structures that achieve psychological, aesthetic and environmental aspects in an integrated design. Therefore, Self- Similarity Dimension (Ds) Box-counting Dimension (Db.) All of these dimensions are directly related to the fractional dimension of Mandelbrot (D). In all similar constructions there is a relationship between the scale factor and the number of the smaller pieces the original construction is divided into.

On the relations between antitrace, row/column sum of elements and generalized horadam sequence associated with a matrix

Using Cayley-Hamilton theorem for a given matrix A = [aij] ∈ M2(K), (K = ℝ or ℂ), we obtain the recurrence relations for Tra(An), suRi(An) and suCi(An); Tra(An+2) = Tr(A)Tra(An+1) – det(A) Tra(An) suRi(An+2) = Tr(A)suRi(An+1) – det(A)suRi(An) suCi(An+2) = Tr(A)suCi(An+1) – det(A) suCi(An) where Tra(A) = a12 + a21 is the sum of antidiagonal entries of A and suRi(A) or scCu(A) is the ith(i = 1, 2) row or column sum of A respectively. In doing so, the analogy between the above natural recurrences and the Horadam sequence is observed. Motivated by this, we define the Horadam sequence {Wn(a, b : Tr(A), det(A))} and Fibonacci sequence {Fn = Wn (0, 1; Tr(A), det(A))} associated with A. In this paper, Tra(An), {suRi(An)}, {suCi(An)}, {Wn} and {Fn} sequences are defined and their inter-relationships are extensively studied. We also obtain Binet’s formulae, generating functions, Cassini identities, Vajda’s identities and Catalan’s identities for these sequences. Further, identities are illustrated for particular values of trace and determinant of a matrix