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Summary: Using the familiar way of representing a square of an integer as the sum of consecutive odd integers, the author described a method of studying the Pythagorean triples. This method was used by Fibonnacci in the early 13th century. It is a fruitful and accessible way to explore the following types of problems: given positive integers x2 and m, can there exists a Pythagorean triple x2, y2, z2 such that z = y + m ? Using this method, the author shows that, among other things, there cannot be any primitive Pythagorean triples with m an even positive power of 2, but there are infinitely many such triples with m an odd power of 2. On the other hand, if p is an odd prime, there cannot be any primitive triples with m an odd power of p, but infinitely many such triples with m an even power of p. To encourage students to use this method, the author has also designed a sequence of problems for students to explore. [ABSTRACT FROM AUTHOR]

In this paper, the Fibonacci sequence, renowned for its significance across various fields, its ability to illuminate numerical concepts, and its role in uncovering patterns in mathematics and nature, forms the foundation of this research. This study introduces innovative concepts of weighted density, weighted statistical summability, weighted statistical convergence, and weighted statistical Cauchy, uniquely defined via the Fibonacci sequence and modulus functions. Key theorems, relationships, examples, and properties substantiate these novel principles, advancing our comprehension of sequence behavior. Additionally, we extend the notion of statistical cluster points within a broader framework, surpassing traditional definitions and offering deeper insights into convergence in a statistical context. Our findings in this paper open avenues for new applications and further exploration in various mathematical fields. [ABSTRACT FROM AUTHOR]

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence { f n } n ∈ N 0 having initial terms f 0 = 0 , and f 1 = 1 and recurrence relation f n = f n − 1 + f n − 2 (n ≥ 2) . In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension t f n ∗ , with initial terms t f 0 ∗ = (f 0 ∗ , f 1 ∗ , f 2 ∗) and t f 1 ∗ = (f 1 ∗ , f 2 ∗ , f 3 ∗) , which is generated by the recurrence relation t f n ∗ = t f n − 1 ∗ + t f n − 2 ∗ (n ≥ 2) , the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities. [ABSTRACT FROM AUTHOR]

Firstly studied by Kempa and Prezza in 2018 as the unifying idea behind text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we focus on string attractors of prefixes of particular automatic infinite words (including the famous period-doubling and k-bonacci words) related to simple-Parry numbers. For a subfamily of these words, we describe string attractors of optimal size, while for the rest of them, we provide nearly optimal-size ones. Such a contribution is of particular interest, since in general finding smallest string attractors is NP-hard. This extends our previous work published in the international conference WORDS 2023. [ABSTRACT FROM AUTHOR]

Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled (1 × n) and (2 × n) square grids by covering them with squares and dominoes. We describe these numbers not only recursively, but also as rational polynomial linear combinations of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

This communication, based on a review of the relevant literature on ratios deriving from blood pressure and heart rate measurements, and their conformance/nonconformance to the mathematical golden rule (ie, 1.681), proposes that such ratios, particularly emanating from large numbers of home blood pressure and heart rate measurements obtained by the patients themselves or their caretakers, may constitute new risk markers, useful in the assessment of health and cardiovascular pathologies, prognosis of morbidity and mortality, and implementation to clinical practice and research. [ABSTRACT FROM AUTHOR]

Resonant acoustic band gap materials have steered a new sensing technology era. This study is presented to investigate of the one-dimensional (1D) phononic crystals (PnCs), involving periodic, as well as quasi-periodic 1D layered PnCs represented as a highly sensitive biosensor to detect and monitor the quality of milk. In this regard, the numerical findings show that the examined periodic PnCs structure outperformed the quasi-periodic structure. In particular, it provides a wider phononic band gap and greater sensitivity as well. In addition, the quasi-periodic design (especially Fibonacci sequence S4) introduces multiple resonance peaks via transmission spectra, which may lead to some conflicts during the detection process. The findings reveal that the frequency of the resonant peak can effectively change with varied milk solution concentrations and temperatures. The optimized sensor is capable of differentiating between concentrations ranging between 0 and 50 % with a 10 % step, and it can also differentiate between temperatures, which range between 5 °C and 50 °C. This makes it ideal for precise detection of other liquids and solutions. The sensor performs efficiently for all milk solution concentrations. Here, the findings demonstrated that the examined defective PnC structure exhibited the most favorable sensitivity of the value of 94.34 MHz, so it showed the highest sensitivity when varying milk concentrations. In addition, the configurated sensor provided high QF and FOM values of 3,853.645161 and 157.42, respectively. On the other hand, the sensor performs very well for all temperatures of the milk solution. As such, the S4 quasi-periodic structure is characterized as the optimal sensor structure when varying temperatures, introducing a sensitivity of 4.78 MHz/°C, QF of 4,278.521, and FOM of 7.48 °C−1. [ABSTRACT FROM AUTHOR]

Computational and technological advancements have led to an increase in data generation and storage capacity. Many annotated datasets have been used to train machine learning models for predictive tasks. Feature selection (FS) is a combinatorial binary optimization problem that arises from a need to reduce dataset dimensionality by finding the subset of features with maximum predictive accuracy. While different methodologies have been proposed, metaheuristics adapted to binary optimization have proven to be reliable and efficient techniques for FS. This paper applies the first and unique population-trajectory metaheuristic, the Lichtenberg algorithm (LA), and enhances it with a Fibonacci sequence to improve its exploration capabilities in FS. Substituting the random scales that controls the Lichtenberg figures' size and the population distribution in the original version by a sequence based on the golden ratio, a new optimal exploration–exploitation LF's size decay is presented. The new few hyperparameters golden Lichtenberg algorithm (GLA), LA, and eight other popular metaheuristics are then equipped with the v-shaped transfer function and associated with the K-nearest neighbor classifier in the search of the optimized feature subsets through a double cross-validation experiment method on 15 UCI machine learning repository datasets. The binary GLA selected reduced subsets of features, leading to the best predictive accuracy and fitness values at the lowest computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a comprehensive theoretical study to obtain an optimal highly sensitive fluidic sensor based on the one-dimensional phononic crystal (PnC). The mainstay of this study strongly depends on the high impedance mismatching due to the irregularity of the considered quasi-periodic structure, which in turn can provide better performance compared to the periodic PnC designs. In this regard, we performed the detection and monitoring of the different concentrations of lead nitrate (Pb(NO3)2) and identified it as being a dangerous aqueous solution. Here, a defect layer was introduced through the designed structure to be filled with the Pb(NO3)2 solution. Therefore, a resonant mode was formed within the transmittance spectrum of the considered structure, which in turn shifted due to the changes in the concentration of the detected analyte. The numerical findings demonstrate the role of the different sequences such as Fibonacci, Octonacci, Thue–Morse, and double period on the performance of the designed PhC detector. Meanwhile, the findings of this study show that the double-period quasi-periodic sequence provides the best performance with a sensitivity of 502.6 Hz/ppm, a damping rate of 5.9 × 10 − 5 , a maximum quality factor of 8463.5, and a detection limit of 2.45. [ABSTRACT FROM AUTHOR]

This work presents a new public key cryptography encryption-decryption scheme based on generalized Fibonacci matrices and the Hill cipher. This scheme proposes using Fibonacci sequences under prime modulo for key establishment. This scheme exchanges the key matrix X = Mθp of order p × p for encryption and decryption. As an alternative to a key matrix, our scheme only requires the exchange of a pair of numbers (p,θ), resulting in a small key space and reduced time and space complexity. We also implemented our scheme with the help of java script and established better security.

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.

We extend the Kannan contraction principle and obtain a result that holds for both contractive and non-expansive mappings. Such mappings admit multiple fixed-points and the fixed-point sets and domains of these mappings display interesting algebraic, geometric and dynamical features. Since contraction mappings admit only one fixed-point, almost all the existing results on contractive mappings can be generalized in the light of our theorem. As an application of our main theorem, we obtain the integral solutions of a nonlinear Diophantine equation; the solutions are Pythagorean triples, which represent right angled triangles, and each integer of the triple belongs to a Fibonacci type sequence. These results can be generalised to obtain integral solutions of Diophantine equations of the type (n+k)2 – n2 = p2, k > 1, and to check whether the related sequences are Fibonacci sequences.

AbstractWe examine the equation for positive integers y, a ≥ 2 and k ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y ≤ 25000 when k = 3, for y ≤ 1000 when k = 4, for y ≤ 40 when k = 5, and for y ≤ 3 when k = 6. [ABSTRACT FROM AUTHOR]

We investigate optimal expansions of Kakeya sequences for the representation of real numbers. Expansions of Kakeya sequences generalize the expansions in non-integer bases and they display analogous redundancy phenomena. In this paper, we characterize optimal expansions of Kakeya sequences, and we provide conditions for the existence of unique expansions with respect to Kakeya sequences. [ABSTRACT FROM AUTHOR]

In this study, we investigate the vertices arising from the action of a suborbital graph, in terms of continued fractions, matrix, and recurrence relations. Using the approximation of Fibonacci sequence by the Binet formula, we demonstrate that the vertices of the suborbital graph are related to Lucas numbers. Then, we provide new identities and approximations regarding Fibonacci, Lucas, Pell, and Pell-Lucas numbers. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate the following three symmetric Toeplitz determinants: $$\begin{align*} &{\rm det}{[|j-k|+\delta_{jk}]_{1\leq j,k\leq n}},\ {\rm det}[F_{|j-k|}+\delta_{jk}]_{1\le j,k\le n} \quad {\rm and}\\ &{\rm det}\left[\genfrac{(}{)}{}{}{|j-k|}{3}-\delta_{jk}\right]_{1\le j,k\le n}, \end{align*} $$ det [ | j − k | + δ jk ] 1 ≤ j , k ≤ n , det [ F | j − k | + δ jk ] 1 ≤ j , k ≤ n and det [ ( | j − k | 3 ) − δ jk ] 1 ≤ j , k ≤ n , where $ \delta _{jk} $ δ jk is the Kronecker delta, $ (F_i)_{i\ge 0} $ (F i) i ≥ 0 is the Fibonacci sequence, and $ \left (\frac {\cdot }3\right) $ (⋅ 3) is the Legendre symbol. [ABSTRACT FROM AUTHOR]

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths. [ABSTRACT FROM AUTHOR]

Summary: Wireless body area network (WBAN) is a potential low‐cost technology for privacy‐sensitive telemedicine and e‐health monitoring and services. However, it faces limited protocol and physical resource support challenges, which can result in packet transfer difficulties. In particular, WBAN requires an emergency‐aware technology that ensures a promising quality of service (QoS). One significant issue affecting QoS and energy efficiency in WBAN is congestion. Effective congestion control techniques are essential for achieving proper load balancing. To address these challenges, we propose a congestion severity aware rate control (CSRR) algorithm that enhances packet transmission rate by reducing packet losses and retransmissions. The CSRR algorithm incorporates a fuzzy controller to predict congestion rates based on runtime metrics. To regulate congestion window growth in different algorithm phases, we introduce sequences such as the Fibonacci retracement sequence, knight's move sequence, and the binary logarithm of the Nth primorial sequence to regulate congestion window growth in the different phases of the proposed algorithm. We mathematically analyze the proposed CSRR algorithm using a Markov model. The simulation results demonstrate the superiority of our algorithm compared to existing approaches. Specifically, our algorithm achieves significant optimizations in terms of throughput (52.92%), packet loss (38.11%), delay (37.23%), and remaining energy (36.86%) when compared to existing algorithms. [ABSTRACT FROM AUTHOR]

The combinatorial properties of Fibonacci words and arrays are now a popularly studied concept in formal language theory. One such property is the palindromic property of words and arrays. In this paper, we construct algorithms to generate the involutive Fibonacci words and involutive Fibonacci arrays and to check, if a given word is involutive Fibonacci or not. We verify the algorithms using python programming. And we investigate the number of palindromes present in the sequences of involutive Fibonacci words and involutive Fibonacci arrays which has various applications. [ABSTRACT FROM AUTHOR]

Compared with periodic structures, quasi-periodic structures have superior band gap properties and topological interface states. In this paper, a one-dimensional quasi-periodic Fibonacci water wave metamaterial model that can be used to apply quasi-periodic structures to shallow-water wave systems is presented. The fluctuation characteristics of periodic and quasi-periodic structures are examined using finite element numerical calculations based on the shallow-water wave equation. The research results show that the band characteristics of quasi-periodic structures are complex, enabling flexible control of the propagation of shallow-water waves. Furthermore, the mirror-symmetrical design of Fibonacci quasi-periodic water wave metamaterials was created to engineer the topological interface states in shallow-water wave systems, ultimately achieving successful localization of wave energy. This research will greatly enrich our understanding of topology, expand the potential applications of quasi-periodic structures, and provide new insights for manipulating water waves and harvesting energy. [ABSTRACT FROM AUTHOR]

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In this paper, we give an analytical demonstration of the possibility to realize a simple photonic multichannel tunable filter based on electromagnetically induced transparency (EIT) resonances. On one hand, we present a simple photonic device consisting of multiple grafted resonators at the same site; the resonator lengths depend on each other following the Fibonacci sequence. We have proposed this simple device to obtain EIT type resonances that are situated between two transmission zeros because each resonator induces its own transmission zero. The quality factor of these EIT resonances depends on the difference between the resonator lengths. On the other hand, we investigate the propagation of electromagnetic waves in one-dimensional perfect star waveguides (SWGs) structure composed of the periodicity of segments and grafted in its extremity by a finite number of symmetric resonators. This perfect star waveguide structure presents passbands separated by large bandgaps, these gaps become wider when the number of resonators increases. The insertion of five defect resonators of lengths d02i = d02 (i = 1–5) located in the same site introduce transmission peaks (defect mode) in the transmission and phase spectrums and in the phase with a high-quality factor (very narrow defect modes) that reach Q = 1 348 700 and therefore our proposed case behaves as a very narrow filter which can select one frequency. To realize a multichannel filter with high performance, we introduce five defect resonators of different lengths d02i (i = 1–5) according to the Fibonacci sequence in the perfect SWGs. These defect systems can create a large number of defect modes which reach seven defect modes with an important transmission rate and very high-quality factor. With an appropriate choice of the geometrical parameters (the lengths d02i of the defect resonators), our proposed system can filter a very large number of frequencies (multichannel filter) which can reach ninety-nine frequencies with very high performances. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article focuses on characterizing a class of quasi-periodic metamaterials created through the repeated arrangement of an elementary cell in a fixed direction. The elementary cell consists of two building blocks made of elastic materials and arranged according to the generalized Fibonacci sequence, giving rise to a quasi-periodic finite microstructure, also called Fibonacci generation. By exploiting the transfer matrix method, the frequency band structure of selected periodic approximants associated with the Fibonacci superlattice, i.e. the layered quasi-periodic metamaterial, is determined. The self-similarity of the frequency band structure is analysed by means of the invariants of the symplectic transfer matrix as well as the transmission coefficients of the finite clusters of Fibonacci generations. A high-frequency continualization scheme is then proposed to identify integral-type or gradient-type non-local continua. The frequency band structures obtained from the continualization scheme are compared with those derived from the Floquet–Bloch theory to validate the proposed scheme. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1).' [ABSTRACT FROM AUTHOR]

A customized metamaterial tailored for a specific functionality holds significant appeal in practical applications, yet its alteration after the structure is established can be challenging. A novel design for Fibonacci‐array inspired acoustic metamaterials is introduced, which are constructed using metamaterial bricks with unique physical mechanisms. This design aims to achieve multifunctional low‐frequency sound absorption. The Fibonacci sequence arrangement flexibly modulates the coupling between metamaterial bricks, thereby improving energy‐dissipating efficiency. Additionally, the strategic alignment enhances the wave‐absorbing properties of the metamaterial, allowing it to demonstrate remarkable absorption effects across targeted frequency bands. By controlling the resonance effect of metamaterial bricks in intensive and sparse modes, the proposed design exhibited frequency‐selective performance, resulting in three absorption peaks at 323, 687, and 1113 Hz, respectively, across low‐ to high‐frequency ranges. Furthermore, the broadband absorption performance, characterized by strong coupling strength, enables continuous sound absorption over a low‐frequency band from 290 to 440 Hz. This is supported by theoretical analysis, numerical simulations, and experimental results, showcasing the flexible modulation of the propagation characteristics of sound waves. Overall, this functionally actuated design dramatically enhances the tunability of the metamaterials and offers a promising avenue for multifunctional application in noise‐control engineering. [ABSTRACT FROM AUTHOR]

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related 2 × 2 matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated 2 × 2 matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this study, we examine the modulation of light transmission via a one-dimensional polymeric symmetrical Fibonacci-multilayer. The multilayer is constructed using substitutional generalised Fibonacci using PMMA and PS as the constituent materials. The widely recognised transfer matrix approach is utilised to derive theoretical results. The results of our investigation indicate that the width, spacing, and depth of the PBG can be modified by choosing the proper generation number. Remarkably, the count of significant PBGs remains consistent throughout all generation numbers, with the exception of the 10 th generation symmetrical Fibonacci-multilayer, which exhibits a pronounced localization. As the generation number grows, a symmetrical shift of the PBGs is noticed in relation to their position. Moreover, it has been shown that the forbidden frequency range can be adjusted by changing the hydrostatic pressure applied. The rise in pressure causes a shift in the frequency spectrum towards higher frequencies. Furthermore, our findings indicate that augmenting the polymer thickness leads to a shift towards longer wavelengths, known as a redshift, in the PBGs. This paper presents a new method for creating optical devices, specifically multi-band tunable optical filters and durable optical sensors, using one-dimensional polymeric symmetrical Fibonacci multilayers. [ABSTRACT FROM AUTHOR]

The given text contains a collection of mathematical formulas, equations, and graphs related to various topics such as the calculation of pi, the Fibonacci sequence, and the arcsine function. The text provides specific formulas and calculations, as well as proofs for some of the formulas. [Extracted from the article]

In this study, we examined the Oresme sequences defined by Nicole Oresme. We examined the geometric interpretation of Oresme sequences with rational coefficients which are defined by A.F. Horadam as 0n+2= 0n+1- 1/40nwith initial conditions and 00 = 0 and 01 = 1/2. We defined the the vector of the Oresme sequence. We calculated the area and volume. We gave the general solution for four squares equation involving Oresme vectors. We calculated the Heron Formula of Oresme sequences. We defined the angle value between these sequences. We also obtained a relationship between the Oresme sequence and the generalized Fibonacci sequence in vector space. We calculated the area and volume of these sequence. We obtained important definitions and theorems for these sequences. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ -Fibonacci statistical convergence, strong Δ -Fibonacci summability, and Δ -Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers. [ABSTRACT FROM AUTHOR]

In this research, we proposed a new concept called as the H -Nacci sequence. The H -Nacci sequence (Fibonacci sequences of length h) is a collection of numbers developed from the coefficients of the generalized m-th Fibonacci equation. After that, we determined the golden ratio for each type of H -Nacci sequence, which also coincided with an existing Fibonacci sequence. As each Fibonacci sequence has a unique advantage, first of all, we have applied the H -Nacci sequence to the virus mutation process to show the key benefits of the H -Nacci sequence, and then we found the Fibonacci risk model to analyze the risk factor of each mutant virus using the H -Nacci sequence. [ABSTRACT FROM AUTHOR]

Recently Bilu, Marques and Togbé [4] gave a general effective finiteness result on the equation F n (k) = C m , where F n (k) denotes the k -generalized Fibonacci-sequence and C m the sequence of Cullen numbers, by giving explicit absolute bounds for n , k , m. However, the authors in [4] explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in [4] and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers. [ABSTRACT FROM AUTHOR]

IT IS KNOWN THAT THE FIBONACCI SEQUENCE IS USED IN DOZENS OF FIELDS SUCH AS FINANCE, NATURE, ART, CRYPTOGRAPHY, MUSIC AND ECONOMY. THEY ARE FINDING THE (N)TH ELEMENT IN THE FIBONACCI SEQUENCE REQUIRES CALCULATING THE (N-1) AND (N-2) ELEMENTS. THE FIRST AND SECOND ELEMENTS ARE CALCULATED RECURSIVELY. THE FIBONACCI SEQUENCE IN THIS STUDY IS ASCERTAINED USING THE PASCAL TRIANGLE. A FORMULA FOR DIRECTLY DETERMINING THE NECESSARY FIBONACCI ELEMENT HAS BEEN PROPOSED. WHEN ALL OF THE ELEMENTS IN THE DIAGONAL PLANE ARE GATHERED IN PASCAL TRIANGLES FROM LEFT TO RIGHT, IT IS KNOWN THAT THE ELEMENTS OF THE SEQUENCE KNOWN AS FIBONACCI CAN BE CALCULATED SEQUENTIALLY. THE HIDDEN PATTERN WITHIN THIS TRIANGLE IS CONVERTED INTO A NEW FORMULA BY UTILIZING MATHEMATICAL FUNCTIONS, SIGMA SYMBOLS, AND COMBINATION MODELING. RECURSION AND DYNAMIC PROGRAMMING COMPUTATIONS YIELD THE MINIMAL TIME AND SPACE COMPLEXITY TO FIND THE FIBONACCI SERIES. [ABSTRACT FROM AUTHOR]

In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana's sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal's triangle. [ABSTRACT FROM AUTHOR]

The text is a summary of the Proceedings of the 17th International Workshop on Real and Complex Singularities, which took place in 2022. The workshop is a renowned event in singularity theory and related fields, bringing together experts and young researchers to share their achievements and discuss research trends. The proceedings contain 29 research papers that have been selected and refereed according to the standards of the journal Research in the Mathematical Sciences. The collection aims to provide readers, including graduate students and researchers, with an opportunity to explore the field of singularities. The text also highlights the contributions and achievements of Professor Osamu Saeki, a leading researcher in singularity theory and topology, who was honored during the workshop. [Extracted from the article]

We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show that the sequence of the network's equivalent resistance converges uniformly in the parameter α = r 2 r 1 ∈ [ 0 , + ∞) , where r 1 and r 2 are the first and second resistors in the network. We also show that these networks exhibit self-similarity and scale invariance, which mimics a self-similar fractal. We also provide some generalizations, including resistor networks based on high-order Fibonacci sequences and other recursive combinatorial sequences. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We extend the Kannan contraction principle and obtain a result that holds for both contractive and non-expansive mappings. Such mappings admit multiple fixed-points and the fixed-point sets and domains of these mappings display interesting algebraic, geometric and dynamical features. Since contraction mappings admit only one fixed-point, almost all the existing results on contractive mappings can be generalized in the light of our theorem. As an application of our main theorem, we obtain the integral solutions of a nonlinear Diophantine equation; the solutions are Pythagorean triples, which represent right angled triangles, and each integer of the triple belongs to a Fibonacci type sequence. These results can be generalised to obtain integral solutions of Diophantine equations of the type (n+k)² - n² = p², k > 1, and to check whether the related sequences are Fibonacci sequences. [ABSTRACT FROM AUTHOR]

A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely $(a,b)$-Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly.

In this article, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. Next, we show the main advantage of our formula which is based on the fact that future calculations rely on the previous ones and this is true from any desired starting point. As applications, we show that Binet’s formula, in this case, is valid for negative integers as well. Finally, new summation formulas relating elements of such sequences. As a conclusion, we present a formula for the sum of squares of Chebychev polynomials of the first and second kind.

In this study, we define the (d, k)−Fibonacci polynomial and examine its properties. We give the generating function, characteristic equation, and matrix representation of this polynomial. Then we get the infinite sum for the (d, k)−Fibonacci polynomials. We give the relationship between (d, k)−Fibonacci polynomial and d− Fibonacci polynomial. Also, with the help of (d, k)−Fibonacci polynomial matrix representation and the Riordan matrix, the factorization of the Pascal matrix in two different ways is given. In addition, we define the infinite (d, k)− Fibonacci polynomial matrix and give their inverses. The Riordan arrays linked here help us understand patterns of number concepts and prove many theorems, as well as helping us make an intuitive connection for solving combinatorial problems. Among our main goals is to combine Riordan arrays with the Fibonacci number sequence, which is the most important of the number sequences, and to expand this study to the k-Fibonacci number sequence, which is the general form of Fibonacci number sequences. Based on the information given above, Riordan array and Pascal matrices, which have an important place in matrix theory and combinatorics studies also it derived an encoding of Pascal’s triangle in matrix form, were discussed in this study and a very different generalization of the Fibonacci number sequence was studied. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence, a special number sequence studied a lot recently and plays an important role with its applications in many fields of science, can be obtained in different areas of mathematics and with different methods. In this study, Fibonacci numbers are obtained with using suborbital graphs of the Modular group Г and some special matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new generalization of Pascal's triangle, which we call the bi-periodic Pascal's triangle. We establish the recurrence relation for the sequences obtained from the sums of elements lying on finite rays in this triangle and derive the corresponding generating function. Additionally, we study the bi-periodic Morgan-Voyce phenomenon, present several combinatorial identities, and provide a new explicit formula for the classical Morgan-Voyce sequence. Finally, we give a formula for the partial sums of such sequences. [ABSTRACT FROM AUTHOR]