Fibonacci Numbers Research Articles

On the Fibonacci Tiling and its Modern Ramifications

AbstractIn the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well‐known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher‐dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

Fibonacci Numbers between History, Semiotics, and Storytelling: The Birth of Recursive Thinking

The aim of this paper is to discuss the emergence of recursive thinking through the famous problem posed by Fibonacci regarding the growth of the rabbit population. This paper qualitatively analyzes and discusses the semiotic aspects raised by the students working with this historical source in the form of a story. From this perspective, the value of the historical problems as socio-cultural references (voices) and of the narrations as mediating factors to enhance students’ learning of new mathematical concepts, such as recursion, is explored in depth. The focus lies on the pivotal role played by the students’ construction of personal senses during in-group mathematical activities, in dialectics with the normative and mathematical meanings. It is highlighted that fostering environments conducive to dialogue among peers, as well as linking various shapes and contexts of knowledge, is necessary. Here, storytelling and history are regarded as fruitful resources aiding students in the gradual construction of a personal sense of mathematical concepts, including recursion.

On the Hankel matrices of finite rank and generalized Fibonacci sequences

On the Hankel matrices of finite rank and generalized Fibonacci sequences

Abstract CT166: First-in-human phase 1 study of a phosphatidylserine-targeting small molecule drug conjugate (T-1201) in patients with advanced solid tumors

Abstract Background Phosphatidylserine (PS), a phospholipid, is an essential component of bilayer cell membranes and is normally present in the inner cell membrane layer of the cells. In rapid proliferating cancer cells, PS externalizes to the outer cell membrane and has the potential to act as a tumor biomarker and therapeutic target. T-1201 is a small molecule drug conjugate comprised of a phosphatidylserine-targeting head, Zinc (II) bis-dipicolylamine (Zn-DPA), linked to bioactive topoisomerase I inhibitor SN-38. Once T-1201 via Zn-DPA targets tumor cells, then enzymatic cleavage releases the cytotoxic payload of SN38, the active metabolite of irinotecan. This leads to high accumulation of antineoplastic drug directly in the tumor microenvironment. Here, we describe a trial in progress of a first-in-human trial of T-1201 in patients with advanced solid tumors. Methods This first-in-human, open-label, multicenter, phase I trial (NCT04866641) is investigating the safety, tolerability, pharmacokinetics and antitumor activity of T-1201 administrated as monotherapy to subjects with advanced solid tumors. Eligible patients have ≥ 1 measurable lesion per RECIST v1.1. and no effective standard therapy exists. T-1201 is administered intravenously once every 4 weeks (Q4W) in 28-day cycles with dose escalation guided by an accelerated titration design. The study started with 1 subject enrolled per cohort. With 100% dose increment for the next cohort. If 1 subject experiences any ≥ Grade 2 AE or DLT at any dose level during first cycle, the dose escalation will transit to 3+3 design where dose increment will follow a modified Fibonacci sequence. Starting with Cycle 2, the dosing interval can be adjusted to once every two weeks (Q2W) and the dose level after switching to Q2W schedule will be half of that at Q4W. Intra-subject dose escalation is permitted to maximize the likelihood of responses while keeping toxicity acceptable for individual subjects. The dose can be increased after subjects complete at least 2 treatment cycles at a particular dose level without experiencing any > grade 2 drug-related toxicity. The primary endpoints are the maximum tolerated dose (MTD), recommended Phase 2 dose (RP2D) and adverse event rate as measured by CTCAE v5.0. Secondary endpoints include objective response rate (ORR), clinical benefit rate (CBR), duration of response (DOR) and time to tumor progression. This study has begun enrolling patients and is continuing as planned in Taiwan. 77% of planned patients have been enrolled as of November 2023. Citation Format: Hui-Ching Wang, Yu-Min Yeh, Jeng-Shiun Du, Tsung-Jang Yeh, Jui-Hung Tsai, Yu-Chin Gau, Shang Yin Wu, Li-Tzong Chen, Hsiao-Fang Li, Wu-Chou Su. First-in-human phase 1 study of a phosphatidylserine-targeting small molecule drug conjugate (T-1201) in patients with advanced solid tumors [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 2 (Late-Breaking, Clinical Trial, and Invited Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(7_Suppl):Abstract nr CT166.

Enhancement of superconductivity in the Fibonacci chain

Enhancement of superconductivity in the Fibonacci chain

Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board

The tradition of studies involving the combinatorial approach to recurring numerical sequences has accumulated a few decades of tradition, and several problems continue to attract the interest of mathematicians in several countries. This work specifically discusses the Fibonacci, Pell, and Jacobsthal sequences, focusing on Mersenne sequences. The often-used definition of board involves considering how to fill a specific regular surface -the board- with a limited quantity of regularly shaped tiles. On the other hand, an analogous problem can be generalized and exemplifies current research developments. Finally, the examples covered constitute unexpected ways of exploring visualization and other skills in mathematics teachers’ learning, consequently inspiring them for their teaching context.

Global Behavior of Solutions to a Higher-Dimensional System of Difference Equations with Lucas Numbers Coefficients

In this paper, we derive the well-defined solutions to a θ-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results.

Generalization of the Fibonacci sequence, Pascal's triangle, and the binomial theorem

هدف هذا البحث إلى تعميم العلاقة بين مثلث باسكال وتوسيع ثنائي باستخدام المتغيرات بدلاً من الأرقام. يتم تشكيل المثلث باستخدام المتغير (d) بدلاً من المصطلح الصفري (0)، والمتغير (a) بدلاً من المصطلح الأول (1)، والمتغير (m) كتعميم لنظرية الثنائي. يتم دراسة الأنماط الرياضية الناتجة عن تشكيل المثلث باستخدام هذه المتغيرات، مما يؤدي إلى خمسة معادلات رياضية جديدة: معادلة عمودية، معادلة الوتر، معادلة الصف، مجموع معادلات الصفوف، ومعادلة تسلسلات الوظائف الذهبية. تُعتبر معادلة تسلسلات الوظائف الذهبية تعميمًا غير مسبوق للمصطلح الثاني لتسلسل فيبوناتشي وتسلسل لوكاس. بالإضافة إلى ذلك، يُصاغ معادلة جديدة وغير مسبوقة للماس، ويُصاغ افتراض جديد يتعلق بأعداد أولية، حيث يُعتبر تعميمًا لنظرية فيرما الصغرى. يُسلط هذا البحث الضوء على ضرورة فهم أكثر شمولًا للعلاقة بين مثلث باسكال وتوسيع ثنائي.

Bidiagonal Factorizations of Filbert and Lilbert Matrices

Extensions of Filbert and Lilbert matrices are addressed in this work. They are reciprocal Hankel matrices based on Fibonacci and Lucas numbers, respectively, and both are related to Hilbert matrices. The Neville elimination is applied to provide explicit expressions for their bidiagonal factorization. As a byproduct, formulae for the determinants of these matrices are obtained. Finally, numerical experiments show that several algebraic problems involving these matrices can be solved with outstanding accuracy, in contrast with traditional approaches.

Leonardo and hyper-Leonardo numbers via Riordan arrays

UDC 511 A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A - and Z -sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are obtained using the fundamental theorem of the Riordan arrays.

C++ and Kotlin performance on Android – a comparative analysis

The article discusses the comparison of C++ and Kotlin programming languages in a mobile environment. The authors performed a series of tests based on five selected algorithms: n-bodies, the nth term of the Fibonacci sequence, reading and writing a file, and bubble sort for both small and large sets of values. The tests were carried out in a way that allowed to determine the performance of the Kotlin language both when it uses the Just-in-Time compilation mechanism and when it is not used. The research was carried out both on a physical mobile device and emulators. Although the C⁠+⁠+ language outclassed its rival in most of the tests performed, Kotlin showed more than three times faster performance when bubble sorting on a small (20,000 values) array.

Fekete-Szegö type functionals associated with certain subclasses of bi-univalent functions

In this article, we define and discuss some new subclasses of bi-univalent functions associated with Fibonacci numbers and shell-like curves. We estimate the first two Taylor-Maclaurin coefficients for functions belonging to these classes. We study the inequalities related with Fekete-Szegö type. Furthermore, we point out some corollaries related to existing result.

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.

Generation of dynamic rotation propagation vortex beam by a Fibonacci series annular subzone vortex phase

Abstract Recently, the optical vortex (OV) has gained increasing interest since the potential for applications of the orbital angular momentum (OAM) carried by optical vortex beams. However, generation is currently limited single static cricular intensity profile, greatly constraining the breadth of achievable spatiotemporal dynamics. Here, we propose a novel phase distribution to generate a dynamic propagation OV with a customized topological charge (TC ≥ 10) based on Fibonacci series annular subzone by tailoring the local phase gradient along the azimuthal direction. We describe the generation of the Fibonacci series annular subzone vortex phase. The Fibonacci vortex beam (FVB) have customization TC, multi-singularity intensity distributions. Furthermore, such optical fields exhibit the spatial dynamic rotation and self-focusing have yielded fascinating phenomena. The simulation results are agreed with the experimental results, which provides an important basis for the generation of OV with spatial dynamic propagation. These results contribute to the advanced complex light manipulation with spatial dynamic propagation and pave the way to achieve a new laser with the structured light based on modified phase control.