Fibonacci-like Sequences Research Articles

On Solving Third, Fourth and Fifth Order of Reverse Fibonacci-Like Sequences

On Solving Third, Fourth and Fifth Order of Reverse Fibonacci-Like Sequences

A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity

Abstract Filbert and Lilbert matrices are defined by terms of linear Fibonacci-like sequences. By considering terms of such linear recurrences and additional free parameters, we define a nonlinear variant of these matrices via ratios of q-forms of terms of Fibonacci-like sequences whose indices are in nonlinear forms. We derive explicit formulae for the matrices L and U come from LU-decomposition, their inverses, inverse of the main matrices and their determinants.

Fibonacci-like Sequences Reveal the Genetic Code Symmetries, Also When the Amino Acids Are in a Physiological Environment

In this study, we once again use a set of Fibonacci-like sequences to examine the symmetries within the genetic code. This time, our focus is on the physiological state of the amino acids, considering them as charged, in contrast to our previous work where they were seen as neutral. In a pH environment around 7.4, there are four charged amino acids. We utilize the properties of our sequences to accurately describe the symmetries in the genetic code table. These include Rumer’s symmetry, the third-base symmetry and the “ideal” symmetry, along with the “supersymmetry” classification schemes. We also explore the special chemical structure of the amino acid proline, presenting two perspectives—shCherbak’s view and the Downes–Richardson view—which are included in the description of the above-mentioned symmetries. Our investigation also employs elementary modular arithmetic to precisely describe the chemical structure of proline, connecting the two views seamlessly. Finally, our Fibonacci-like sequences prove instrumental in quickly establishing the multiplet structure of non-standard versions of the genetic code. We illustrate this with an example, showcasing the efficiency of our method in unraveling the complex relationships within the genetic code.

Revealing the Genetic Code Symmetries through Computations Involving Fibonacci-like Sequences and Their Properties

In this work, we present a new way of studying the mathematical structure of the genetic code. This study relies on the use of mathematical computations involving five Fibonacci-like sequences; a few of their “seeds” or “initial conditions” are chosen according to the chemical and physical data of the three amino acids serine, arginine and leucine, playing a prominent role in a recent symmetry classification scheme of the genetic code. It appears that these mathematical sequences, of the same kind as the famous Fibonacci series, apart from their usual recurrence relations, are highly intertwined by many useful linear relationships. Using these sequences and also various sums or linear combinations of them, we derive several physical and chemical quantities of interest, such as the number of total coding codons, 61, obeying various degeneracy patterns, the detailed number of H/CNOS atoms and the integer molecular mass (or nucleon number), in the side chains of the coded amino acids and also in various degeneracy patterns, in agreement with those described in the literature. We also discover, as a by-product, an accurate description of the very chemical structure of the four ribonucleotides uridine monophosphate (UMP), cytidine monophosphate (CMP), adenosine monophosphate (AMP) and guanosine monophosphate (GMP), the building blocks of RNA whose groupings, in three units, constitute the triplet codons. In summary, we find a full mathematical and chemical connection with the “ideal sextet’s classification scheme”, which we alluded to above, as well as with others—notably, the Findley–Findley–McGlynn and Rumer’s symmetrical classifications.

Fibonacci-like Sequences for Variants of the Tower of Hanoi, with Corresponding Graphs and Gray Codes

Fibonacci-like Sequences for Variants of the Tower of Hanoi, with Corresponding Graphs and Gray Codes

On d-Fibonacci digraphs

The d -Fibonacci digraphs F ( d , k ) , introduced here, have the number of vertices following some generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, F ( d , k ) has diameter d + k − 2 and is semi-pancyclic; that is, it has a cycle of every length between 1 and ℓ , with ℓ ∈ {2 k − 2, 2 k − 1} . Moreover, it turns out that several other numbers of F ( d , k ) (of closed l -walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the d -Fibonacci digraphs.

Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

<p style='text-indent:20px;'>We consider three matrix models of order 2 with one random entry <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the other three entries being deterministic. In the first model, we let <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $\end{document}</tex-math></inline-formula>. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when <inline-formula><tex-math id="M3">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(p\right) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p\in [0, 1] $\end{document}</tex-math></inline-formula> is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with <inline-formula><tex-math id="M5">\begin{document}$\xi\epsilon$\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$\epsilon$\end{document}</tex-math></inline-formula> is a standard Cauchy random variable and <inline-formula><tex-math id="M7">\begin{document}$\xi$\end{document}</tex-math></inline-formula> is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

Generalized (p,q)-Fibonacci-Like Sequences and Their Properties

In this paper, we define the generalized (p,q) -Fibonacci-Like sequences Sp,q,n &amp;nbsp;associated with the (p,q) -Fibonacci and the (p,q) -Lucas sequences and then we get some fundamental identities and sums formulas involving odd and even terms of the generalized (p,q) -Fibonacci-Like sequences. We obtain the Binet&amp;#39;s formula to find the n th general term of generalized (p,q) -Fibonacci-Like sequences Sp,q,n . Also, the generating functions of these sequences are presented and proved.In this paper, we define the generalized (p,q) -Fibonacci-Like sequences Sp,q,n &amp;nbsp;associated with the (p,q) -Fibonacci and the (p,q) -Lucas sequences and then we get some fundamental identities and sums formulas involving odd and even terms of the generalized (p,q) -Fibonacci-Like sequences. We obtain the Binet&amp;#39;s formula to find the n th general term of generalized (p,q) -Fibonacci-Like sequences Sp,q,n . Also, the generating functions of these sequences are presented and proved.

Organized structures of two bidirectionally coupled logistic maps.

We report some organized structures of two linearly coupled logistic maps with different harvesting. The coupled system exhibits chaos via period-bubbling and quasiperiodic routes for identical and weak coupling strength, in contrast to conventional period-doubling route for a simple logistic map. Studies reveal the existence of infinite families of periodic Arnold tongues and self-similar shrimp-shaped structures with period-adding sequences for periodic windows embedded in quasiperiodic and chaotic regions, respectively. Different Fibonacci-like sequences are formed leading to the Golden Mean. The shrimp-shaped structures maintain period 3-times self-similarity scaling. The quasiperiodicity route is the necessary condition for the occurrence of periodic Arnold tongues in this coupled system resulting in the appearance of shrimps in the chaotic region near the tongues. It is also revealed that the existence of shrimp implies the period-bubbling cascade but the reverse is not true. The bifurcation-induced hysteresis is born in a certain parameter range resulting in the birth of coexisting multiple attractors of different kinds. Basin sets of the coexisting attractors have either self-similar or intertwining fractal basin boundaries.

Matrix representations of Fibonacci-like sequences

Matrix representations of Fibonacci-like sequences

Relative growth of the partial sums of certain random Fibonacci-like sequences

We consider certain Fibonacci-like sequences perturbed with a random noise. Our main result is that converges in distribution, as n goes to infinity, to a random variable W with Pareto-like distribution tails. We show that is a monotonically decreasing characteristic of the input noise, and hence can serve as a measure of its strength in the model. Heuristically, the heavy-tailed limiting distribution, versus a light-tailed one with can be interpreted as an evidence supporting the idea that the noise is ‘singular’ in the sense that it is ‘big’ even in a ‘slightly’ perturbed sequence.

Periodicity of Generalized Fibonacci-like Sequences

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence F_(n+2)=A+BF_(n+1)+CF_n. It is shown that this sequence is periodic with the period T>2 if C=-1,|B|<2.

Fibonacci-Like sequence associated with k-Pell, k-Pell-Lucas and Modified k-Pell sequences

The main aim of the present article to introduce a generalization of Fibonacci sequence which is similar to k-Pell, k-Pell-Lucas, Modified k-Pell sequences and known as Fibonacci-Like sequence. After that we obtain some fundamental properties of Fibonacci-Like sequence such as Binet formulae of Fibonacci-Like sequence, binomial transform of the Fibonacci-Like sequence and some other properties of Fibonacci-Like sequence with indexes in an arithmetic sequence. In addition to this we obtain some relations among k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences and the most worth noticing cause of the present paper is we present determinantal identities involving k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences.

Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, , and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Bn=Bn-1+Bn-2, with B0=2s, B1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

Identities of Generalized Fibonacci-Like Sequence

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, n≥2 and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Mn=Mn-1+Mn-2, n≥2, with M0=2, M1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence are presented by Binet’s formula. Also some determinant identities are discussed.

Tribonacci-like sequences and generalized Pascal's pyramids

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences.

On convergence properties of Fibonacci-like conditional sequences

On convergence properties of Fibonacci-like conditional sequences

Fibonacci-like sequences and generalized Pascal’s triangles

The properties pertaining to diagonals of generalized Pascal’s triangles are studied. Combinatorial relationships between Fibonacci-like sequences and Fibonacci sequence itself are determined, using the sequence of diagonals of generalized Pascal’s triangle.

Generalized Identities of Companion Fibonacci-Like Sequences

The Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, Jacobsthalsequence and Jacobsthal-Lucas sequence are most prominent examples of second order recursivesequences. In this paper, we deal with two companion Fibonacci- Like sequences which aregeneralization of Fibonacci-Like sequence. Further we obtain some generalized identities amongthe terms of companion Fibonacci-Like sequences, Jacobsthal and Jacobsthal-Lucas sequencesthrough Binet’s formulae.

Pisot [formula omitted]-coherent states quantization of the harmonic oscillator

We revisit the quantized version of the harmonic oscillator obtained through a q-dependent family of coherent states. For each q, 0<q<1, these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which q−1 is a quadratic unit Pisot number, since then the q-deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, angle operator, probability distributions and related statistical features, time evolution and semi-classical phase space trajectories.