Fibonacci Number
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On K-Fibonacci Numbers of Arithmetic Indexes
Applied Mathematics and Computation 208 (2009) 180–185 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On k-Fibonacci numbers of arithmetic indexes Sergio Falcon *, Angel Plaza Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), Campus de Tafira, 35017 Las Palmas de Gran Canaria, Spain article info abstract Keywords: In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic k-Fibonacci numbers sequence, say an þ r for fixed integers a and r. This enables us to give in a straightforward Sequences of partial sums way several formulas for the sums of such numbers. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction One of the more studied sequences is the Fibonacci sequence [1–3], and it has been generalized in many ways [4–10]. Here, we use the following one-parameter generalization of the Fibonacci sequence. Definition 1. For any integer number k P 1, the kth Fibonacci sequence, say fFk;ngn2N is defined recurrently by Fk;0 ¼ 0; Fk;1 ¼ 1; and Fk;nþ1 ¼ kFk;n þ Fk;nÀ1 for n P 1: Note that for k ¼ 1 the classical Fibonacci sequence is obtained while for k ¼ 2 we obtain the Pell sequence. Some of the properties that the k-Fibonacci numbers verify and that we will need later are summarized below [11–15]: pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi n n 2 2 r1Àr2 kþ k þ4 kÀ k þ4 [Binet’s formula] Fk;n ¼ r Àr , where r1 ¼ 2 and r2 ¼ 2 . These roots verify r1 þ r2 ¼ k, and r1 Á r2 ¼1 1 2 2 nþ1Àr 2 [Catalan’s identity] Fk;nÀrFk;nþr À Fk;n ¼ðÀ1Þ Fk;r 2 n [Simson’s identity] Fk;nÀ1Fk;nþ1 À Fk;n ¼ðÀ1Þ n [D’Ocagne’s identity] Fk;mFk;nþ1 À Fk;mþ1Fk;n ¼ðÀ1Þ Fk;mÀn [Convolution Product] Fk;nþm ¼ Fk;nþ1Fk;m þ Fk;nFk;mÀ1 In this paper, we study different sums of k-Fibonacci numbers. -
From a GOLDEN RECTANGLE to GOLDEN QUADRILATERALS And
An example of constructive defining: TechSpace From a GOLDEN TechSpace RECTANGLE to GOLDEN QUADRILATERALS and Beyond Part 1 MICHAEL DE VILLIERS here appears to be a persistent belief in mathematical textbooks and mathematics teaching that good practice (mostly; see footnote1) involves first Tproviding students with a concise definition of a concept before examples of the concept and its properties are further explored (mostly deductively, but sometimes experimentally as well). Typically, a definition is first provided as follows: Parallelogram: A parallelogram is a quadrilateral with half • turn symmetry. (Please see endnotes for some comments on this definition.) 1 n The number e = limn 1 + = 2.71828 ... • →∞ ( n) Function: A function f from a set A to a set B is a relation • from A to B that satisfies the following conditions: (1) for each element a in A, there is an element b in B such that <a, b> is in the relation; (2) if <a, b> and <a, c> are in the relation, then b = c. 1It is not being claimed here that all textbooks and teaching practices follow the approach outlined here as there are some school textbooks such as Serra (2008) that seriously attempt to actively involve students in defining and classifying triangles and quadrilaterals themselves. Also in most introductory calculus courses nowadays, for example, some graphical and numerical approaches are used before introducing a formal limit definition of differentiation as a tangent to the curve of a function or for determining its instantaneous rate of change at a particular point. Keywords: constructive defining; golden rectangle; golden rhombus; golden parallelogram 64 At Right Angles | Vol. -
On the Prime Number Subset of the Fibonacci Numbers
Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach On the Prime Number Subset of the Fibonacci Numbers Lacey Fish1 Brandon Reid2 Argen West3 1Department of Mathematics Louisiana State University Baton Rouge, LA 2Department of Mathematics University of Alabama Tuscaloosa, AL 3Department of Mathematics University of Louisiana Lafayette Lafayette, LA Lacey Fish, Brandon Reid, Argen West SMILE Presentations, 2010 LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions History Two Famous and Useful Sieves Sieve of Eratosthenes Brun’s Sieve Lacey Fish, Brandon Reid, Argen West -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. -
Twelve Simple Algorithms to Compute Fibonacci Numbers Arxiv
Twelve Simple Algorithms to Compute Fibonacci Numbers Ali Dasdan KD Consulting Saratoga, CA, USA [email protected] April 16, 2018 Abstract The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used in introductory algorithms courses. In this paper, we present twelve of these well-known algo- rithms and some of their properties. These algorithms, though very simple, illustrate multiple concepts from the algorithms field, so we highlight them. We also present the results of a small-scale experi- mental comparison of their runtimes on a personal laptop. Finally, we provide a list of homework questions for the students. We hope that this paper can serve as a useful resource for the students learning the basics of algorithms. arXiv:1803.07199v2 [cs.DS] 13 Apr 2018 1 Introduction The Fibonacci numbers are a sequence Fn of integers in which every num- ber after the first two, 0 and 1, is the sum of the two preceding num- bers: 0; 1; 1; 2; 3; 5; 8; 13; 21; ::. More formally, they are defined by the re- currence relation Fn = Fn−1 + Fn−2, n ≥ 2 with the base values F0 = 0 and F1 = 1 [1, 5, 7, 8]. 1 The formal definition of this sequence directly maps to an algorithm to compute the nth Fibonacci number Fn. However, there are many other ways of computing the nth Fibonacci number. -
Thesis Final Copy V11
“VIENS A LA MAISON" MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz A Thesis Submitted to the Faculty of The Dorothy F. Schmidt College of Arts & Letters in Partial Fulfillment of the Requirements for the Degree of Master of Art in Teaching Art Florida Atlantic University Boca Raton, Florida May 2011 "VIENS A LA MAlSO " MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz This thesis was prepared under the direction of the candidate's thesis advisor, Angela Dieosola, Department of Visual Arts and Art History, and has been approved by the members of her supervisory committee. It was submitted to the faculty ofthc Dorothy F. Schmidt College of Arts and Letters and was accepted in partial fulfillment of the requirements for the degree ofMaster ofArts in Teaching Art. SUPERVISORY COMMIITEE: • ~~ Angela Dicosola, M.F.A. Thesis Advisor 13nw..Le~ Bonnie Seeman, M.F.A. !lu.oa.twJ4..,;" ffi.wrv Susannah Louise Brown, Ph.D. Linda Johnson, M.F.A. Chair, Department of Visual Arts and Art History .-dJh; -ZLQ_~ Manjunath Pendakur, Ph.D. Dean, Dorothy F. Schmidt College ofArts & Letters 4"jz.v" 'ZP// Date Dean. Graduate Collcj;Ze ii ACKNOWLEDGEMENTS I would like to thank the members of my committee, Professor John McCoy, Dr. Susannah Louise Brown, Professor Bonnie Seeman, and a special thanks to my committee chair, Professor Angela Dicosola. Your tireless support and wise counsel was invaluable in the realization of this thesis documentation. Thank you for your guidance, inspiration, motivation, support, and friendship throughout this process. To Karen Feller, Dr. Stephen E. Thompson, Helena Levine and my colleagues at Donna Klein Jewish Academy High School for providing support, encouragement and for always inspiring me to be the best art teacher I could be. -
Page 1 Golden Ratio Project When to Use This Project
Golden Ratio Project When to use this project: Golden Ratio artwork can be used with the study of Ratios, Patterns, Fibonacci, or Second degree equation solutions and with pattern practice, notions of approaching a limit, marketing, review of long division, review of rational and irrational numbers, introduction to ϕ . Appropriate for students in 6th through 12th grades. Vocabulary and concepts Fibonacci pattern Leonardo Da Pisa = Fibonacci = son of Bonacci Golden ratio Golden spiral Golden triangle Phi, ϕ Motivation Through the investigation of the Fibonacci sequence students will delve into ratio, the notion of irrational numbers, long division review, rational numbers, pleasing proportions, solutions to second degree equations, the fascinating mathematics of ϕ , and more. Introductory concepts In the 13th century, an Italian mathematician named Leonardo Da Pisa (also known as Fibonacci -- son of Bonacci) described an interesting pattern of numbers. The sequence was this; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Notice that given the first two numbers, the remaining sequence is the sum of the two previous elements. This pattern has been found to be in growth structures, plant branchings, musical chords, and many other surprising realms. As the Fibonacci sequence progresses, the ratio of one number to its proceeding number is about 1.6. Actually, the further along the sequence that one continues, this ratio approaches 1.618033988749895 and more. This is a very interesting number called by the Greek letter phi ϕ . Early Greek artists and philosophers judged that a page 1 desirable proportion in Greek buildings should be width = ϕ times height. The Parthenon is one example of buildings that exhibit this proportion. -
Fibonacci Is Not a Pizza Topping Bevel Cut April, 2020 Andrew Davis Last Month I Saw an Amazing Advertisement Appearing in Our Woodworking Universe
Fibonacci is Not a Pizza Topping Bevel Cut April, 2020 Andrew Davis Last month I saw an amazing advertisement appearing in our woodworking universe. You might say that about all of the Woodpeckers advertisements – promoting good looking, even sexy tools and accessories for the woodworker who has money to burn or who does some odd task many times per day. I don’t fall into either category, though I often click on the ads to see the details and watch a video or two. These days I seem to have time to burn, if not money. Anyway, as someone who took too many math courses in my formative years, this ad absolutely caught my attention. I was left guessing two things: what is a Fibonacci and why would anyone need such a tool? Mr. Fibonacci and his Numbers Mathematician Leonardo Fibonacci was born in Pisa – now Italy – in 1170 and is credited with the first CNC machine and plunge router. Fibonacci popularized the Hindu–Arabic numeral system in the Western World (thereby replacing Roman numerals) primarily through his publication in 1202 called the Book of Calculation. Think how hard it would be to do any woodworking today if dimensions were called out as XVI by XXIV (16 x 24). But for this bevel cut article, the point is that he introduced Europe to the sequence of Fibonacci numbers. Fibonacci numbers are a sequence in which each number is the sum of the two preceding numbers. Here are the first Fibonacci numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418. -
In This Research Report We Will Explore the Gestalt Principles and Their Implications and How Human’S Perception Can Be Tricked
The Gestalt Principles and there role in the effectiveness of Optical Illusions. By Brendan Mc Kinney Abstract Illusion are created in human perception in relation to how the mind process information, in this regard on to speculate that the Gestalt Principles are a key process in the success of optical illusions. To understand this principle the research paper will examine several optical illusions in the hopes that they exhibit similar traits used in the Gestalt Principles. In this research report we will explore the Gestalt Principles and their implications and how human’s perception can be tricked. The Gestalt Principles are the guiding principles of perception developed from testing on perception and how human beings perceive their surroundings. Human perception can however be tricked by understanding the Gestalt principles and using them to fool the human perception. The goal of this paper is to ask how illusions can be created to fool human’s perception using the gestalt principles as a basis for human’s perception. To examine the supposed, effect the Gestalt Principles in illusions we will look at three, the first being Rubin’s Vase, followed by the Penrose Stairs and the Kanizsa Triangle to understand the Gestalt Principles in play. In this context we will be looking at Optical Illusion rather than illusions using sound to understand the Gestalt Principles influence on human’s perception of reality. Illusions are described as a perception of something that is inconsistent with the actual reality (dictionary.com, 2015). How the human mind examines the world around them can be different from the actuality before them, this is due to the Gestalt principles influencing people’s perception. -
The Psychological Intersection of Motion Picture, the Still Frame, and Three-Dimensional Form
MOMENTUM, MOMENT, EPIPHANY: THE PSYCHOLOGICAL INTERSECTION OF MOTION PICTURE, THE STILL FRAME, AND THREE-DIMENSIONAL FORM by MARK GERSTEIN B.A. The University of Chicago, 1986 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Fine Arts in the School of Visual Arts and Design in the College of Arts and Humanities at the University of Central Florida Orlando, Florida Spring Term 2018 ABSTRACT My journey from Hollywood Film production to a Fine Arts practice has been shaped by theory from Philosophy of Mind, Cognitive Psychology, Film, and Art, leading me to a new visual vocabulary at the intersection of motion picture, the still image, and three-dimensional form. I create large mixed media collages by projecting video onto photographs and sculptural forms, breaking the boundaries of the conventional film frame and exceeding the dynamic range of typical visual experience. My work explores emotional connections and fissures within family, and hidden meanings of haunting memories and familiar places. I am searching for an elusive type of perceptual experience characterized by an instantaneous shift in perspective—an “aha” moment of epiphany when suddenly I have the overpowering feeling that I am both seeing and aware that I am seeing. ii To Lori, Joshua and Maya, for your infinite patience and unconditional love. iii ACKNOWLEDGMENTS I want to take this opportunity to acknowledge all my Studio Art colleagues who have so graciously tolerated my presence in their sandbox over these last few years. In particular, I want to thank my Thesis Committee: Carla Poindexter, my chair, for her nuanced critique, encouragement and unwavering belief in the potential of my work, and ability to embrace the seemingly contradictory roles of mentor and colleague; Jo Anne Adams, for her attention to detail and narrative sensibility that comes from our shared background in the film industry; and Ryan Buyssens, for showing me the possibilities of technology and interactivity, and reminding me to never lose sight of the meaning in my art. -
Exploring the Golden Section with Twenty-First Century Tools: Geogebra
Exploring the Golden Section with Twenty-First Century Tools: GeoGebra José N. Contreras Ball State University, Muncie, IN, USA [email protected] Armando M. Martínez-Cruz California State University, Fullerton, CA, USA [email protected] ABSTRACT: In this paper we illustrate how learners can discover and explore some geometric figures that embed the golden section using GeoGebra. First, we introduce the problem of dividing a given segment into the golden section. Second, we present a method to solve said problem. Next, we explore properties of the golden rectangle, golden triangle, golden spiral, and golden pentagon. We conclude by suggesting some references to find more appearances of the golden section not only in mathematics, but also in nature and art. KEYWORDS: Golden section, golden rectangle, golden triangle, golden spiral, golden pentagon, GeoGebra. 1. Introduction Interactive geometry software such as GeoGebra (GG) allows users and learners to construct effortlessly dynamic diagrams that they can continuously transform. The use of such software facilitates the teaching and learning of properties of mathematical objects, such as numbers and geometric figures. One of the most ubiquitous numbers is the so called golden number, denoted by the Greek letter φ (phi) in honor to Phidias who used it in the construction of the Parthenon in Athens. The golden number is involved in the solution to the following geometric problem: 퐴퐵 퐴푃 Given a segment ̅퐴퐵̅̅̅, find an interior point P such that = (Fig. 1). In other words, point 퐴푃 푃퐵 P divides segment ̅퐴퐵̅̅̅, into two segments (̅퐴퐵̅̅̅ and 푃퐵̅̅̅̅ ) such that the ratio of the entire segment to the larger segment is equal to the ratio of the larger segment to the smaller segment. -
Lionel March Palladio's Villa Emo: the Golden Proportion Hypothesis Rebutted
Lionel Palladio’s Villa Emo: The Golden Proportion March Hypothesis Rebutted In a most thoughtful and persuasive paper Rachel Fletcher comes close to convincing that Palladio may well have made use of the ‘golden section’, or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. What is surprising is that a visually gratifying result is so very wrong when tested by the numbers. Lionel March provides an arithmetic analysis of the dimensions provided by Palladio in the Quattro libri to reach new conclusions about Palladio’s design process. Not all that tempts your wand’ring eyes And heedless hearts, is lawful prize; Nor all that glisters, gold (Thomas Gray, Ode on the Death of a Favourite Cat) Historical grounding In a most thoughtful and persuasive paper [Fletcher 2000], Rachel Fletcher comes close to convincing that Palladio may well have made use of the ‘golden section’, or extreme and mean ratio, in the design of the Villa Emo at Fanzolo which was probably conceived and built during the decade 1555-1565. It is early in this period, 1556, that I dieci libri dell’archittetura di M. Vitruvio Pollionis traduitti et commentati ... by Daniele Barbaro was published by Francesco Marcolini in Venice and the collaboration of Palladio acknowledged. In the later Latin edition [Barbaro 1567], there are geometrical diagrams of the equilateral triangle, square and hexagon which evoke ratios involving 2 and 3, but there are no drawings of pentagons, or decagons, which might explicitly alert the perceptive reader to the extreme and mean proportion, 1 : I :: I : I2.