Relevant bibliographies by topics / Pascal's triangel

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Pascal's triangel'

Author: Grafiati
Published: 4 June 2021
Last updated: 5 February 2022

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Journal articles on the topic "Pascal's triangel"

1

Brothers, Harlan J. "Pascal's Prism." Mathematical Gazette 96, no. 536 (2012): 213–20. http://dx.doi.org/10.1017/s0025557200004447.

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Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:
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Matsui, H., D. Minematsu, T. Yamauchi, and R. Miy Adera. "Pascal-like triangles and Fibonacci-like sequences." Mathematical Gazette 94, no. 529 (2010): 27–41. http://dx.doi.org/10.1017/s0025557200007129.

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In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.
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Németh, László. "Fibonacci words in hyperbolic Pascal triangles." Acta Universitatis Sapientiae, Mathematica 9, no. 2 (2017): 336–47. http://dx.doi.org/10.1515/ausm-2017-0025.

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Abstract The hyperbolic Pascal triangle HPT4,q (q ≥ 5) is a new mathematical construction, which is a geometrical generalization of Pascal’s arithmetical triangle. In the present study we show that a natural pattern of rows of HPT 4,5 is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a HPT 4,q imply a graph structure between the finite Fibonacci words.
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Lewis, Barry. "Generalising Pascal's Triangle." Mathematical Gazette 88, no. 513 (2004): 447–56. http://dx.doi.org/10.1017/s0025557200176089.

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Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.
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Howard, Marilyn. "Discovering Patterns in Pascal's Triangle." Mathematics Teaching in the Middle School 24, no. 4 (2019): 247–54. http://dx.doi.org/10.5951/mathteacmiddscho.24.4.0247.

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The less you know about the patterns in Pascal's triangle, the more fun you will have discovering the triangle's many secrets. I am amazed at how few students and even teachers (especially at the middle school level) have ever explored Pascal's triangle. Although this famous triangle bears the name of Blaise Pascal (1623-1662), who saw many of the patterns when he was only thirteen years old, it had been around for centuries before he was born. See the ancient diagram in figure 1, which appeared at the front of a Chinese book in 1303 (Vakil 2008). Evidence suggests that the properties of the elements of Pascal's triangle were known before the common era. Students and teachers alike can enjoy exploring patterns through problem solving with Pascal's triangle.
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Moghaddamfar, A. R. "DETERMINANTS OF SEVERAL MATRICES ASSOCIATED WITH PASCAL'S TRIANGLE." Asian-European Journal of Mathematics 03, no. 01 (2010): 119–31. http://dx.doi.org/10.1142/s1793557110000088.

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Pascal's triangle is one of the most well-known arithmetical triangles and has many wonderful properties. This triangle may be rearranged so that one can consider various matrices. When these matrices are squares, we can discuss their determinants. Our purpose of this article is to study the determinants of square matrices related to Pascal's triangle where the determinants are equal to an entry in a particular place. We also consider the square matrices whose determinants are related to their dimensions only.
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Putz, John F. "The Pascal Polytope: An Extension of Pascal's Triangle toNDimensions." College Mathematics Journal 17, no. 2 (1986): 144–55. http://dx.doi.org/10.1080/07468342.1986.11972945.

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Solares-Hernández, Pedro A., Fernando A. Manzano, Francisco J. Pérez-Benito, and J. Alberto Conejero. "Divisibility Patterns within Pascal Divisibility Networks." Mathematics 8, no. 2 (2020): 254. http://dx.doi.org/10.3390/math8020254.

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The Pascal triangle is so simple and rich that it has always attracted the interest of professional and amateur mathematicians. Their coefficients satisfy a myriad of properties. Inspired by the work of Shekatkar et al., we study the divisibility patterns within the elements of the Pascal triangle, through its decomposition into Pascal’s matrices, from the perspective of network science. Applying Kolmogorov–Smirnov test, we determine that the degree distribution of the resulting network follows a power-law distribution. We also study degrees, global and local clustering coefficients, stretching graph, averaged path length and the mixing assortative.
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Shcherban, V. "Arithmetic Table as an Integral Part of all Computational Mathematics." Bulletin of Science and Practice 6, no. 6 (2020): 31–41. http://dx.doi.org/10.33619/2414-2948/55/04.

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The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.
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Ferreira, Luis Dias. "Arithmetic Triangle." Journal of Mathematics Research 9, no. 2 (2017): 100. http://dx.doi.org/10.5539/jmr.v9n2p100.

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The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.
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Dissertations / Theses on the topic "Pascal's triangel"

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Davila, Rosa. "Tribonacci Convolution Triangle." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/883.

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A lot has been said about the Fibonacci Convolution Triangle, but not much has been said about the Tribonacci Convolution Triangle. There are a few ways to generate the Fibonacci Convolution Triangle. Proven through generating functions, Koshy has discovered the Fibonacci Convolution Triangle in Pascal's Triangle, Pell numbers, and even Tribonacci numbers. The goal of this project is to find inspiration in the Fibonacci Convolution Triangle to prove patterns that we observe in the Tribonacci Convolution Triangle. We start this by bringing in all the information that will be useful in constructing and solving these convolution triangles and find a way to prove them in an easy way.
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Saucedo, Antonio Jr. "Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/855.

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Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.
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James, Lacey Taylor. "Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/835.

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This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the coefficients of the binomial expansion. Furthermore, combinatorics plays an increasingly important role in mathematics, starting when students enter high school and continuing on into their college years. Students become familiar with the traditional arguments based on classical arithmetic and algebra, however, methods of combinatorics can vary greatly. In combinatorics, perhaps the most important concept revolves around the coefficients of the binomial expansion, studying and proving their properties, and conveying a greater insight into mathematics.
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Silva, MÃrcio RebouÃas da. "NÃmeros binomiais: uma abordagem combinatÃria para o ensino mÃdio." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15115.

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Este trabalho tem por finalidade apresentar uma abordagem, para o Ensino MÃdio, de nÃmeros binomiais (incluindo as propriedades do triÃngulo de Pascal e binÃmio de Newton), contendo as demonstraÃÃes combinatÃrias, ao utilizar dupla contagem, juntamente com as demonstraÃÃes algÃbricas, como parcialmente jà à feito, alÃm de generalizar, citando os nÃmeros trinomiais (incluindo as propriedades da pirÃmide de Pascal) e os nÃmeros multinomiais (incluindo o polinÃmio de Leibniz).<br>This project aims at presenting an approach of binomial numbers for high school (including Pascalâs triangle properties and binomial of Newton), containing the combinatorial statements when using double counting, along with algebraic demonstrations, as part is already done in addition to generalize, citing the trinomial numbers (including the properties of the Pascal pyramid) and multinomial numbers (including the Leibnizâs polynomial).
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Riderer, Lucia. "Numbers of generators of ideals in local rings and a generalized Pascal's Triangle." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2732.

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This paper defines generalized binomial coefficients and shows that they can be used to generate generalized Pascal's Triangles and have properties analogous to binomial coefficients. It uses the generalized binomial coefficients to compute the Dilworth number and the Sperner number of certain rings.
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Amaca, Edgar Gilbuena. "On rational functions with Golden Ratio as fixed point /." Online version of thesis, 2008. http://hdl.handle.net/1850/6212.

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ROSADAS, VITOR DUTRA SOARES. "PASCAL S TRIANGLE: CURIOSITIES AND APPLICATIONS IN PRIMARY SCHOOL." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2016. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=28192@1.

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Essa dissertação tem como objetivo principal proporcionar novos olhares sobre o Triângulo de Pascal na escola básica. Este é um assunto rico e pouco explorado nesse segmento escolar. Possibilita o desenvolvimento de aplicações e curiosidades interessantes que instigam o interesse do alunado e podem promover um processo tanto de ensino como de aprendizagem mais eficientes. Através de diversas abordagens é possível motivar ou incrementar conteúdos clássicos da Matemática da educação básica além de trabalhar com situações interdisciplinares. O trabalho apresenta um relato histórico do surgimento do Triângulo e seu uso ao longo do tempo por diversos matemáticos até Pascal. São também apresentados e demonstrados resultados matemáticos obtidos a partir da análise dos elementos deste Triângulo. Por fim, uma coletânea de abordagens interessantes que relacionam o Triângulo a diversos campos da Matemática são apresentadas visando possibilitar ao professor da educação básica o uso dessas propostas na criação de atividades da sua sala de aula. Algumas questões com conceitos matemáticos um pouco mais avançados também são explicitadas possibilitando que cada docente escolha e adapte à sua realidade àquelas que julgar pertinentes. Pretende-se assim possibilitar ao professor da escola básica mais um suporte para construção de propostas pedagógicas inovadoras e que contribuam para o desenvolvimento da educação básica de forma mais interessante e mais significativa.<br>This thesis aims to provide new perspectives on Pascal s Triangle in elementary school. This is a rich subject and little explored in this school segment. It enables the development of interesting applications and curiosities that instigate the interest of the students and can promote a process of teaching and learning more effective. Through various approaches-gens can motivate or improve classical mathematics content of basic education as well as working with interdisciplinary situations. The paper presents a historical account of the emergence Triangle and its use over time by several mathematicians to Pascal. They are then presented and demonstrated mathematical results obtained from the analysis of the elements of this triangle. Finally, a collection of interesting approaches that relate the Triangle to various fields of mathematics are presented aiming to enable the primary education teachers use these proposals to create activities of your classroom. Some issues with mathematical concepts a little more advanced are also explicit allowing each teacher choice and adapt to their reality to those it considers relevant. The aim is to enable the teacher of primary school plus a support for building innovative educational proposals and contribute to the development of basic education more interesting and significant.
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Gibbons, Stacie Joyce. "Manipulatives and the Growth of Mathematical Understanding." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3212.

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The purpose of this study was to describe how manipulatives facilitated the growth of one group of high school students' mathematical understanding of combinatorics and Pascal's Triangle. The role of manipulatives in mathematics education has been extensively studied, but much of the interest in manipulatives is focused on the general uses of manipulatives to support student learning. Unfortunately, there is a lack of research that explicitly defines how manipulatives can help students develop mathematical understanding. I have chosen to examine mathematical understanding through the lens of the Pirie-Kieren Theory for Growth of Mathematical Understanding. Through analysis of the students' explorations of the Towers Task, I identified ways in which manipulatives facilitated students' understanding of combinatorics and Pascal's Triangle. It was found that the properties and arrangements of the manipulatives were significant in prompting students' progression through levels of understanding and helped students to reason abstractly and develop mathematical generalizations and theories. From this study we can gain insights into explicit ways in which manipulatives facilitate mathematical understanding. These results have implications for research, teaching and teacher education.
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Aksenov, Alexandre. "Raréfaction dans les suites b-multiplicatives." Phd thesis, Université de Grenoble, 2014. http://tel.archives-ouvertes.fr/tel-00947586.

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On étudie une sous-classe des suites b-multiplicatives rarefiées avec un pas de raréfaction p premier, et on trouve une structure asymptotique avec un exposant alphain]0,1[ et une fonction de raréfaction continue périodique. Cette structure vaut pour les suites qui contiennent des nombres complexes du disque unité (section 1.1), et aussi pour des systèmes de numération avec b chiffres successifs positifs et négatifs (section 1.2). Ce formalisme est analogue à celui décrit (pour le cas particuler de la suite de Thue-Morse) par Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba et autres. Dans la deuxième partie, largement indépendante, on étudie la raréfaction dans les suites composées de -1,0 et +1. On se restreint davantage au cas où b engendre le groupe multiplicatif modulo p. Cette hypothèse est conjecturée (Artin) d'être vraie pour une infinité de nombres premiers. Les constantes qui apparaissent s'expriment alors comme polynômes symétriques des P(zeta^j) où P est un polynôme à coefficients entiers, zeta est une racine primitive p-ième de l'unité, $j$ parcourt les entiers de 1 à p-1 (ce lien est explicité dans la section 1.3). On définit une méthode pour étudier les valeurs de ces polynômes symétriques, basée sur la combinatoire, notamment sur le problème de comptage des solutions des congruences et des systèmes linéaires modulo p avec deux conditions supplémentaires: les résidus modulo p utilisés doivent être non nuls et différents deux à deux. L'importance est donnée à la différence entre les nombres de soluions de deux congruences qui ne diffèrent que du terme sans variable. Le cas des congruences de la forme $x_1+x_2+...+x_n=i mod p$ équivaut à un résultat connu. Le mémoire (section 2.2) lui donne une nouvelle preuve qui en fait une application originale de la formule d'inversion de Möbius dans le p.o.set des partitions d'un ensemble fini. Si au moins deux coefficients distincts sont présents, on peut classer les réponses associées à toutes les congruences possibles qui ont un ensemble fixe de coefficients (de taille d), dans un tableau qu'on va appeler un "simplexe de Pascal fini". Ce tableau est une fonction delta:N^d->Z restreinte aux points de somme des coordonnées inférieure à p (un simplexe), avec deux propriétés: l'équation récursive de Pascal y est vérifiée partout sauf les points où la somme des coefficients est multiple de p (qui seront appelés les "sources" et forment un sous-réseau de l'ensemble des points entiers), et les valeurs en-dehors du simplexe induites par l'équation sont nulles (c'est démontré, en réutilisant la méthode précédente, dans la section 2.3 et en partie 2.4). On décrit un algorithme (section 2.4) qui consiste en applications successives de l'équation dans un ordre précis, qui permet de trouver l'unique fonction delta qui vérifie les deux conditions. On applique ces résultats aux suites b-multiplicatives (dans la section 2.5). On montre aussi que le nombre de sources ne dépend que de la dimension du simplexe d et de la longueur de son côté p. On formule la conjecture (partie 2.6) qu'il serait le plus petit possible parmi les tableaux de forme d'un simplexe de la dimention fixe et taille fixe qui vérifient les mêmes conditions. On montre un premier résultat sur les systèmes de deux congruences linéaires (section 2.5.4), et on montre (section 1.4) un lien avec une méthode de Drmota et Skalba pour prouver l'absence de phénomène de Newman (dans un sens précis), décrit initialement pour la suite de Thue-Morse et tout p tel que b engendre le groupe multiplicatif modulo p, et généralisé (section 1.4) à la suite (-1)^{nombre de chiffres 2 dans l'écriture en base 3 de n} appelée "++-". Cette problématique est riche en problèmes d'algorithmique et de programmation. Différentes sections du mémoire sont illustrées dans l'Annexe. La plupart de ces figures sont inédites.
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Santos, Natânia Laine Paglione. "O misterioso e enigmático mundo de Pascal e Fibonacci." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/152385.

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Submitted by NATÂNIA LAINE PAGLIONE SANTOS null (natania_paglione@hotmail.com) on 2017-12-19T02:02:13Z No. of bitstreams: 1 VERSÃO FINAL PARA ENTREGA - COM AS CAPAS.pdf: 10903941 bytes, checksum: 94b7d3dd00886cba1fe8cdb928889de5 (MD5)<br>Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) Troca da ficha catalográfica, a ficha correta é a elaborada pela Biblioteca. Problema 02) Correção da paginação, da página 06 pula para página 15. Agradecemos a compreensão. on 2017-12-19T11:51:49Z (GMT)<br>Submitted by NATÂNIA LAINE PAGLIONE SANTOS null (natania_paglione@hotmail.com) on 2017-12-23T00:51:17Z No. of bitstreams: 1 DISSERTAÇÃO IMPRESSA E ENCADERNADA.pdf: 11215228 bytes, checksum: b5ff3d316d1fa514f151d233853200ca (MD5)<br>Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-01-02T18:10:17Z (GMT) No. of bitstreams: 1 santos_nlp_me_sjrp.pdf: 11115758 bytes, checksum: de8e1a0afdcaa57b073f0ecf8cdabcfc (MD5)<br>Made available in DSpace on 2018-01-02T18:10:18Z (GMT). No. of bitstreams: 1 santos_nlp_me_sjrp.pdf: 11115758 bytes, checksum: de8e1a0afdcaa57b073f0ecf8cdabcfc (MD5) Previous issue date: 2017-11-09<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Atualmente tem-se percebido uma grande dificuldade em atrelar os conteúdos matemáticos ao cotidiano e estimular os alunos para as aulas. Diante disso percebe-se que demonstrar as fascinantes descobertas do Triângulo de Pascal e a Sequência de Fibonacci ao longo dos anos e suas diversas facetas podem despertar os jovens para um olhar investigativo e curioso, quebrando as barreiras existentes no ensino/aprendizagem de matemática. O objetivo deste estudo foi investigar algumas propriedades e suas demonstrações existentes no Triângulo de Pascal e na Sequência de Fibonacci. Devido ao intrigante assunto escolhido e a pouca exploração nos livros didáticos consultados, abrimos leques de possibilidades para expansão do tema como: Fractais, Sequência de Lucas e Razão Áurea. Para sugestões aos docentes, há na pesquisa aplicações para a sala de aula sobre os temas aqui mencionados, vale ressaltar que o conteúdo relacionado as aplicações da Sequência de Fibonacci e Razão Áurea é espetacular. E como dizia Aristóteles: Os filósofos que afirmam que a Matemática não tem nada a ver com a Estética, estão seguramente errados. A Beleza é de fato o objeto principal do raciocínio e das demonstrações matemáticas.<br>There has been a great difficulty in mathematical content to everyday life and to stimulate students to classrooms. From this we can see that demonstrating the fascinating of the Pascal Triangle and the Fibonacci Sequence to the over the years and its many facets can awaken young people for an investigative and curious look, breaking the barriers in mathematics teaching / learning. The objective of this study was to investigate some properties and their demonstrations in the Pascal Triangle and the Sequence of Fibonacci. Due to the intriguing subject chosen and the few in the textbooks we consulted, we possibilities for expansion of the theme as: Fractais, Sequence of Lucas and Golden Ratio. For suggestions to teachers, there are in the research room applications about the topics mentioned here, it is worth mentioning that the content related to the applications of the Fibonacci Sequence and Golden Ratio is spectacular. And what about Aristotle: 'The philosophers who claim that mathematics has nothing to do with Aesthetics, are surely wrong. THE Beauty is in fact the main object of reasoning and mathematical demonstrations'
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Books on the topic "Pascal's triangel"

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Edwards, A. W. F. Pascal's arithmetical triangle. C. Griffin, 1987.

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Bondarenko, B. A. Obobshchennye treugolʹniki i piramidy Paskali͡a︡, ikh fraktali, grafy i prilozhenii͡a︡. Izd-vo "FAN" Uzbekskoĭ SSR, 1990.

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Decker, Rick. Pascal's triangle: Reading, writing, and reasoning about programs. Wadsworth Pub. Co., 1992.

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Triangular arrays with applications. Oxford University Press, 2011.

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1921-2002, Rousseau Jean, ed. Fermat a-t-il démontré son grand théorème?: L'hypothèse "Pascal" : essai. L'Harmattan, 2002.

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Parallel logic programming. MIT Press, 1991.

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Cai, Zongxi. Deng zhou wen ti. Ke xue chu ban she, 2002.

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Greene, Thomas, Thomas M. Green, and Charles L. Hamberg. Pascal's Triangle-Answer Key. Addison Wesley Publishing Company, 1987.

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Seymour, Dale. Visual Patterns in Pascal's Triangle. Addison Wesley Publishing Company, 1986.

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Korntved, Edward Charles. Greatest common divisor properties in Pascal's triangle. 1991.

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Book chapters on the topic "Pascal's triangel"

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Fulton, Kristy. "Pascal's Triangle." In More Math Puzzles and Patterns For Kids. Routledge, 2021. http://dx.doi.org/10.4324/9781003236733-6.

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Barnes, John. "Pascal’s Triangle." In Nice Numbers. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-46831-0_12.

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Radin, Michael A. "Pascal's Triangle Identities." In Introduction to Math Olympiad Problems. Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003089469-5.

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Radin, Michael A. "Pascal’s Triangle Identities." In Introduction to Recognition and Deciphering of Patterns. Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9780367808747-4.

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Gévay, Gábor. "Pascal’s Triangle of Configurations." In Discrete Geometry and Symmetry. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-78434-2_10.

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Lovász, L., J. Pelikán, and K. Vesztergombi. "Binomial Coefficients and Pascal’s Triangle." In Discrete Mathematics. Springer New York, 2003. http://dx.doi.org/10.1007/0-387-21777-0_3.

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Flath, Dan, and Rhodes Peele. "Hausdorff Dimension in Pascal’s Triangle." In Applications of Fibonacci Numbers. Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2058-6_22.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "Pascal’s Triangle: Cellular Automata and Attractors." In Chaos and Fractals. Springer New York, 2004. http://dx.doi.org/10.1007/0-387-21823-8_9.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "Pascal’s Triangle: Cellular Automata and Attractors." In Chaos and Fractals. Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4740-9_9.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "Pascal’s Triangle: Cellular Automata and Attractors." In Fractals for the Classroom. Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4406-6_3.

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Conference papers on the topic "Pascal's triangel"

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Tomaz, G., M. I. Falcão, and H. R. Malonek. "Pascal's triangle and other number triangles in Clifford analysis." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics. AIP, 2012. http://dx.doi.org/10.1063/1.4756125.

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Chen, Lei, Xinghuo Yu, and Jinhu Lu. "Signal approximation with Pascal’s triangle and sampling." In 2020 Chinese Control And Decision Conference (CCDC). IEEE, 2020. http://dx.doi.org/10.1109/ccdc49329.2020.9164011.

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Tran, MinhTri, Anna Kuwana, and Haruo Kobayashi. "Study of Rauch Low-Pass Filters using Pascal's Triangle." In 2021 International Conference on Electronics, Information, and Communication (ICEIC). IEEE, 2021. http://dx.doi.org/10.1109/iceic51217.2021.9369765.

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Lee, Sangwoo, Jaehoon Choi, and Sunwoo Kim. "Pascal's triangle-based multihop range-free localization for anisotropic sensor networks." In 2014 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2014. http://dx.doi.org/10.1109/wcnc.2014.6952862.

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Zhang, Yunong, Huinan Xiao, Dechao Chen, Sitong Ding, and Pei Chen. "Link from ZD control to pascal's triangle illustrated via multiple-integrator systems." In 2016 12th International Conference on Natural Computation and 13th Fuzzy Systems and Knowledge Discovery (ICNC-FSKD). IEEE, 2016. http://dx.doi.org/10.1109/fskd.2016.7603510.

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Belbachir, Hacène, Takao Komatsu, László Szalay, and Takao Komatsu. "Characterization of linear recurrences associated to rays in Pascal’s triangle." In DIOPHANTINE ANALYSIS AND RELATED FIELDS—2010: DARF—2010. AIP, 2010. http://dx.doi.org/10.1063/1.3478184.

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Belbachir, Hacene, László Németh, and László Szalay. "Properties of hyperbolic Pascal triangles." In INTERNATIONAL CONFERENCE ON MATHEMATICS: PURE, APPLIED AND COMPUTATION: Empowering Engineering using Mathematics. Author(s), 2017. http://dx.doi.org/10.1063/1.4994434.

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Petruccione, F., and I. Sinayskiy. "Open Quantum RandomWalks and the Open Quantum Pascal Triangle." In International Conference on Quantum Information. OSA, 2011. http://dx.doi.org/10.1364/icqi.2011.qmh2.

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Kobayashi, Kingo, Hajime Sato, and Mamoru Hoshi. "The number of paths in boundary restricted Pascal triangle." In 2016 Information Theory and Applications (ITA). IEEE, 2016. http://dx.doi.org/10.1109/ita.2016.7888193.

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Moraga, Claudio, Radomir S. Stankovic, and Milena Stankovic. "The Pascal Triangle (1654), the Reed-Muller-Fourier Transform (1992), and the Discrete Pascal Transform (2005)." In 2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL). IEEE, 2016. http://dx.doi.org/10.1109/ismvl.2016.24.

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