Explore the vast range of titles available.

One of the generalizations of Fibonacci sequence is a -Fibonacci sequence, which is further generalized in several other ways, some by conserving the initial conditions and others by conserving the related recurrence relation. In this paper, we generalize the sequence of -Fibonacci numbers into the sequence of bifurcating Fibonacci numbers. The Binet-like formula for the terms of these numbers is obtained and further, we obtain several interesting properties related to the sequence.

One of the interesting generalizations of Fibonacci sequence is a k-Fibonacci sequence, which is further generalized into the ‘Bifurcating Fibonacci sequence’. In this paper we further generalize it into the sequence of ‘trifurcating Fibonacci numbers’. We obtain the Binet-like formula for these numbers. We also obtain the analogous of Cassini’s identity, Catalan’s identity, d’Ocagne’s identity and some fundamental identities for the terms of this sequence.

In the present study, we introduce a new generalization of Fibonomial coefficients known as bi-periodic Fibonomial coefficients, which can be expressed in relation to bi-periodic Fibonacci numbers. We establish various properties of these coefficients, including recurrence relation and recurrence formulas for powers of the bi-periodic Fibonacci numbers. Moreover, we provide combinatorial interpretation through weighted tilings generated by lattice paths and offer combinatorial proofs for bi-periodic Fibonomial identities.

In this study, we define a new type of number sequence called biperiodic Leonardo sequence by the recurrence relation (for even n ) and (for odd n ) with the initial conditions . We obtained the characteristic function, generating function, and Binet’s formula for this sequence and propose a determinantal representation for the generating function of this sequence. We also provide some properties of these numbers. Moreover, we give the matrix representation of the biperiodic Leonardo numbers.

The golden ratio and the Fibonacci sequence (Fn) are well known, as is the fact that the ratio Fn+1Fn converges to the golden ratio for sufficiently large n. In this paper, we investigate the metallic ratio—a generalized version of the golden ratio—of pulsating Fibonacci sequences in three forms. Two of these forms are considered in the sense of pulsating recurrence relations, and their diagrams can be represented by symmetry, which is one of their distinguishing characteristics. The third form is the Fibonacci sequence in bipolar quantum linear algebra (BQLA), which also pulsates.

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence fnn∈N0 having initial terms f0=0, and f1=1 and recurrence relation fn=fn−1+fn−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 tfn∗, with initial terms tf0∗=(f0∗,f1∗,f2∗) and tf1∗=(f1∗,f2∗,f3∗), which is generated by the recurrence relation tfn∗=tfn−1∗+tfn−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.

The material of this book stems from the idea of integrating a classic concept of Fibonacci numbers with commonly available digital tools including a computer spreadsheet, Maple, Wolfram Alpha, and the graphing calculator. This integration made it possible to introduce a number of new concepts such as: Generalized golden ratios in the form of cycles represented by the strings of real numbers; Fibonacci-like polynomials the roots that define those cycles' dependence on a parameter; the directions of the cycles described in combinatorial terms of permutations with rises, as the parameter changes on the number line; Fibonacci sieves of order k; (r, k)-sections of Fibonacci numbers; and polynomial generalizations of Cassini's, Catalan's, and other identities for Fibonacci numbers. The development of these concepts was motivated by considering the difference equation f_(n+1)=af_n+bf_(n-1),f_0=f_1=1, and, by taking advantage of capabilities of the modern-day digital tools, exploring the behavior of the ratios f_(n+1)/f_n as n increases. The initial use of a spreadsheet can demonstrate that, depending on the values of a and b, the ratios can either be attracted by a number (known as the Golden Ratio in the case a = b = 1) or by the strings of numbers (cycles) of different lengths. In general, difference equations, both linear and non-linear ones serve as mathematical models in radio engineering, communication, and computer architecture research. In mathematics education, commonly available digital tools enable the introduction of mathematical complexity of the behavior of these models to different groups of students through the modern-day combination of argument and computation. The book promotes experimental mathematics techniques which, in the digital age, integrate intuition, insight, the development of mathematical models, conjecturing, and various ways of justification of conjectures. The notion of technology-immune/technology-enabled problem solving is introduced as an educational analogue of the notion of experimental mathematics. In the spirit of John Dewey, the book provides many collateral learning opportunities enabled by experimental mathematics techniques. Likewise, in the spirit of George Plya, the book champions carrying out computer experimentation with mathematical concepts before offering their formal demonstration. The book can be used in secondary mathematics teacher education programs, in undergraduate mathematics courses for students majoring in mathematics, computer science, electrical and mechanical engineering, as well as in other mathematical programs that study difference equations in the broad context of discrete mathematics.

Herein we experimentally study magnetic multilayer metamaterials with broken translational symmetry. Epitaxially-grown iron-gold (Fe-Au) multilayers modulated using Fibonacci sequence—referred to as magnetic inverse Fibonacci-modulated multilayers (IFMs)—are prepared using ultra-high-vacuum vapor deposition. Experimental results of in-situ reflection high-energy electron diffraction, magnetization curves, and ferromagnetic resonance demonstrate that the epitaxially-grown Fe-Au IFMs have quasi-isotropic magnetization, in contrast to the in-plane magnetization easy axis in the periodic multilayers.

In this contribution, we provide a detailed analysis of the search operation for the Interval Merging Binary Tree (IMBT), an efficient data structure proposed earlier to handle typical anomalies in the transmission of data packets. A framework is provided to decide under which conditions IMBT outperforms other data structures typically used in the field, as a function of the statistical characteristics of the commonly occurring anomalies in the arrival of data packets. We use in the modeling Bernstein theorem, Markov property, Fibonacci sequences, bipartite multi-graphs, and contingency tables.