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This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors "Leonardo sequence", "Leonardo numbers", "complexification", "generalization", "hybrids", and "combinatorial model" were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil. [ABSTRACT FROM AUTHOR]

Considering the content of history of mathematics textbooks, it's evident that their emphasis is primarily on the illustrative aspects of recurring numerical sequences, with a particular focus on the Fibonacci sequence. Unfortunately, this limited approach results in the neglect of other sequences akin to the Fibonacci numbers, thus rendering the subject challenging for teaching purposes. This study aims to address this gap by offering a concise exploration of the combinatorial aspects of the Padovan numbers, specifically through the concept of a board as initially examined by mathematicians. In line with the research methodology of didactic engineering and the teaching theory of the theory of didactic situations, two problem situations have been developed, centered on the Padovan combinatorial model, thereby contributing to the enrichment of mathematical education within initial teacher training programs. Within this framework, various strategies are introduced that rely on visualization and counting, with the objective of illustrating specific mathematical identities suitable for potential classroom applications. [ABSTRACT FROM AUTHOR]

Recently, many studies have been devoted to extending particular sets of integers to other special sets, such as complex, dual, hyperbolic, and hybrid numbers. In this study, we define a new generalization of the Leonardo sequence consisting of the hybrid hyper-Leonardo numbers. This sequence of numbers is an extension of the hyper Leonardo numbers in the hybrid set. We investigate some algebraic properties of this sequence, the generating function, and the Binet formula related to this type of sequence. [ABSTRACT FROM AUTHOR]

Figured numbers are sequences of natural numbers that, when represented by dots, form geometric configurations. The immediate sequence of figured numbers presents in its configuration the equilateral triangles, being the basis for finding several others, as long as a relationship between them is discovered. This article shows the construction of polygonal, stellar and three-dimensional figured numbers that can be constructed using the GeoGebra software as a resource to visualize the properties that relate these numbers to the triangular numbers. It is a way of observing the relationship that the figures shown have to each other, how we can understand the structure of these sequences, when presented in the form of figures, and the possibility of building figures that represent some of the most known sequences within the number's figures such as 2D figured numbers such as square, pentagonal, centered triangular and star numbers within 3D figured numbers. Thus, this work presents such construction through the properties of pyramidal numbers, cubic numbers, octahedral numbers and two forms of construction of the stella octangula numbers, with emphasis on the use of triangular numbers or using the other sequences that depart from this one. Furthermore, the work shows the possibility of constructing other sequences as well as the discovery of possible properties that relate them. [ABSTRACT FROM AUTHOR]

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Copyright of Acta Scientiae (1517-4492) is the property of Acta Scientiae and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Acta Scientiae (1517-4492) is the property of Acta Scientiae and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Revista Docência do Ensino Superior is the property of Revista Docencia do Ensino Superior and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Jacobsthal model is a second order recursive sequence. This work presents a discussion on the two-dimensional, three-dimensional and n-dimensional recurrence relations from the one-dimensional recursive model Jn+1 = Jn + 2Jn-1, for all n in N, with J0 = 0 and J1 = 1 being their initial terms. In view of the complexification process of the Jacobsthal sequence, the mathematical properties of the Jacobsthal numbers are considered as J(n,m), J(n, m, p) and J(n1, n2, n3,..., nn), allowing us to explore derived identities inherent in this model. [ABSTRACT FROM AUTHOR]

This work presents a discussion about the Didactic Training Engineering (EDF) methodology, of French origin, emphasizing the role of the teacher, during his initial or continuous training, through the conception and structuring of a didactic situation, organized by such methodology and based on the Theory of Didactic Situations (TSD). Therefore, the objective of this work is to present a didactic situation, based on the research methodology of (EDF) and focused on the analysis of the teacher's role, through the application of the dialectical phases (action, formulation, validation and institutionalization) provided by the TSD. The mathematical problem chosen to compose such a situation, through preliminary and a priori analyzes by EDF, will be focused on Flat Geometry directed to the Permanent Evaluation System of Basic Education in Ceará (SPAECE). SPAECE (Brazil) is a large-scale assessment, held annually in the state of Ceará, Brazil, involving all elementary and high school students in public schools. In addition, the use of the GeoGebra software as a technological resource for carrying out a didactic transposition of the content covered in the proposed didactic situation stands out. For a better conduct of this teaching and learning process, it is suggested the adoption of a didactic contract between the characters involved and the consideration of possible epistemological obstacles and obstacles that may arise during this process. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce the (s; t)-Padovan quaternions matrix sequence. Starting the studies based on the generalization of the Padovan quaternion coefficients in relation to their recurrence, their matrix sequence is then defined. Some mathematical theorems are discussed and the Binet formula and the generating function of this matrix sequence are studied. [ABSTRACT FROM AUTHOR]

The article discusses an alternative way of viewing Padovan sequence extensions through Newton fractals using Google Colab to develop these reproductions. Initially, a scientific bibliographic survey is conducted regarding the process of extension of this sequence. Thus, studies are made on this qualitative alternative of visualization for linear and recurrent sequences. Hereafter alternative forms are developed in Google Colab, so each fractal generated from the characteristic polynomial of these numbers is analyzed. Finally, the conclusions about this process are discussed, proposing studies to continue this work in the future and research in Brazil, as well as possible applications in the areas of Physics and the formation of teachers of Natural Sciences in Brazil. [ABSTRACT FROM AUTHOR]

In Brazil, we are interested in the study of second-order recurrent numerical sequences. Such mathematical content has an enormous potential to provide a differentiated mathematical culture for the math teachers in initial training. However, it becomes important that the teacher in Brazil knows properties beyond the traditional and classic numerical sequences, such as the Fibonacci sequence, Lucas sequence, Pell sequence, Jacobsthal sequence, Narayana sequence, etc. Thus, in the present work, we present a proposal of CAS Maple as technological resources in the study and discovery of new theorems derived from the Mersenne sequence. Throughout the work, we show that the software allows the discovery of theorems from matrix properties derived from the Mersenne sequences and the sequence of Gaussian numbers of Mersenne. The research presented in the paper can stimulate the formation of Mathematics teachers in Brazil. [ABSTRACT FROM AUTHOR]