This thesis is presented as an introduction to the Fibonacci sequence of integers. It is hoped that this thesis will create in the reader more interest in this type of sequence and especially the Fibonacci sequence. It seems that this particular area of mathematics is often ignored in the classroom or touched upon far too briefly to stimulate curiosity and develop further interest in this field.

Combinatorial techniques can frequently provide satisfying “explanations” of various mathematical phenomena. In this thesis, we seek to explain a number of well known number theoretic congruences using combinatorial methods. Many of the results we prove involve the Fibonacci sequence and its generalizations.

O presente trabalho irá abordar dois temas matemáticos de diferentes contextos históricos mas que apresentam uma relação intrínseca com o número Φ, mais conhecido como número de ouro. Partiremos de uma breve descrição dos conjuntos numéricos N, Z, Q e algumas propriedades dos números racionais para, em seguida, deduzirmos os números irracionais Π e, enfim, os números reais R. Na sequência vamos trabalhar com dois problemas muito antigos: o primeiro aparece na coletânea de livros Os Elementos do matemático grego Euclides, 300 anos a.C., e diz respeito à divisão de um segmento em média e extrema razão e, o segundo, foi publicado no livro Liber Abaci do matemático italiano Leonardo Fibonacci, século XIII, e trata da reprodução de coelhos e a sequência a qual ela origina. Veremos que o número de ouro aparece em ambos os problemas e vem ao longo dos séculos desencadeando muitas teorias que tratam de padrões e beleza. Abordaremos situações do passado e do presente que fazem uso desses padrões, além de fenômenos da natureza. Também apresentaremos um conjunto de atividades para orientar professores do ensino médio de como trabalhar, numa perspectiva interdisciplinar com vários conteúdos da matemática, e o número Φ. / This work addresses two mathematical topics from different historical contexts but that have an intrinsic relationship with the number Φ, better known as the golden number. We start with a brief description of the numerical sets N, Z, Q and some properties of rational numbers, and then deduct the set of irrational numbers π and, finally, the set of real numbers R. In the sequence we work with two very old problems: the first appears in the collection of books The elements of the Greek mathematician Euclid, 300 years BC, and concerns the division of a segment in extreme and mean ratio, and the second, published in the book Liber Abaci of the Italian mathematician Leonardo Fibonacci, in the thirteenth century, and deals with the breeding of rabbits and the sequence which it originates. We will see that the golden number appears on both problems and has over the centuries triggering many theories dealing with standards and beauty. We discuss situations of past and present that makes use of these standards, as well as natural phenomena. We also present a set of activities to guide middle school teachers on how to work in an interdisciplinary perspective with various mathematical content, and the number Φ.

O presente trabalho irá abordar dois temas matemáticos de diferentes contextos históricos mas que apresentam uma relação intrínseca com o número Φ, mais conhecido como número de ouro. Partiremos de uma breve descrição dos conjuntos numéricos N, Z, Q e algumas propriedades dos números racionais para, em seguida, deduzirmos os números irracionais Π e, enfim, os números reais R. Na sequência vamos trabalhar com dois problemas muito antigos: o primeiro aparece na coletânea de livros Os Elementos do matemático grego Euclides, 300 anos a.C., e diz respeito à divisão de um segmento em média e extrema razão e, o segundo, foi publicado no livro Liber Abaci do matemático italiano Leonardo Fibonacci, século XIII, e trata da reprodução de coelhos e a sequência a qual ela origina. Veremos que o número de ouro aparece em ambos os problemas e vem ao longo dos séculos desencadeando muitas teorias que tratam de padrões e beleza. Abordaremos situações do passado e do presente que fazem uso desses padrões, além de fenômenos da natureza. Também apresentaremos um conjunto de atividades para orientar professores do ensino médio de como trabalhar, numa perspectiva interdisciplinar com vários conteúdos da matemática, e o número Φ. / This work addresses two mathematical topics from different historical contexts but that have an intrinsic relationship with the number Φ, better known as the golden number. We start with a brief description of the numerical sets N, Z, Q and some properties of rational numbers, and then deduct the set of irrational numbers π and, finally, the set of real numbers R. In the sequence we work with two very old problems: the first appears in the collection of books The elements of the Greek mathematician Euclid, 300 years BC, and concerns the division of a segment in extreme and mean ratio, and the second, published in the book Liber Abaci of the Italian mathematician Leonardo Fibonacci, in the thirteenth century, and deals with the breeding of rabbits and the sequence which it originates. We will see that the golden number appears on both problems and has over the centuries triggering many theories dealing with standards and beauty. We discuss situations of past and present that makes use of these standards, as well as natural phenomena. We also present a set of activities to guide middle school teachers on how to work in an interdisciplinary perspective with various mathematical content, and the number Φ.

No description available.

An n-parameter Fibonacci AF-algebra is determined by a constant incidence matrix K of a special form. The form of the matrix K is defined by a given n-parameter Fibonacci sequence. We compute the K-theory of certain Fibonacci AF-algebra, and relate their K-theory to the K-theory of an AF-algebra defined by incidence matrices that are the transpose of K.

The random Fibonacci sequence is defined by t_1 = t_2 = 1 and t_n = ± t_{n–1} + t_{n–2} , for n ? 3, where each ± sign is chosen at random with P(+) = P(–) = 1/2. We can think of all possible such sequences as forming a binary tree T. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824.... He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. We will discuss a new computation of Viswanath's constant which is based on a formula due to Kalmár-Nagy, and uses an interesting reduction R of the tree T developed by Rittaud. Also, we will focus on the growth rate of the expected value of a random Fibonacci sequence, which was studied by Rittaud. This differs from the almost sure growth rate in that we first find an expression for the average of the n^th term in a sequence, and then calculate its growth. We will derive this growth rate using a slightly different and more simplified method than Rittaud, using the tree R and a Pascal-like array of numbers, for which we can further give an explicit formula. We will also consider what happens to random Fibonacci sequences when we remove the randomness. Specifically, we will choose coefficients which belong to the set {1, –1} and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order, for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length. Lastly we will look at two ways to completely characterize the trace, given the coefficient cycle, by breaking the matrix product up into blocks.

The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given. / Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms.

Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms. / The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given.

Diploma thesis is intended as material for general public. The thesis includes construction methods of the Golden Section and calculation of the Golden Number and its properties. It makes aquainted with the history of the Golden Section and its shows occurrence in a plane geometry. The thesis describes the connection between the Golden Number and the Fibonacci Sequence, occurrence of the Golden Section, Fibonacci Sequnce and the logarithmic spirals in nature in both proportions of living organisms and in plants and on the human body. The text is completed with illustractive figures drawn mostly in GeoGebra. The Golden Number properties and the plane constructions are given. The Golden Section is completed with photos.