Some Links of Balancing and Cobalancing Numbers with Pell and Associated Pell Numbers
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ON SOME GAUSSIAN PELL and PELL-LUCAS NUMBERS Abstract
Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:1, 2016,8-18/Ordu Univ. J. Sci. Tech., Vol:6, No:1,2016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı*1 Sinan Öz2 1Pamukkale Uni., Science and Arts Faculty,Dept. of Math., KınıklıCampus, Denizli, Turkey 2Pamukkale Uni., Science and Arts Faculty,Dept. of Math., Denizli, Turkey Abstract In this study, we consider firstly the generalized Gaussian Fibonacci and Lucas sequences. Then we define the Gaussian Pell and Gaussian Pell-Lucas sequences. We give the generating functions and Binet formulas of Gaussian Pell and Gaussian Pell- Lucas sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Keywords. Recurrence Relation, Fibonacci numbers, Gaussian Pell and Pell-Lucas numbers. Özet Bu çalışmada, önce genelleştirilmiş Gaussian Fibonacci ve Lucas dizilerini dikkate aldık. Sonra, Gaussian Pell ve Gaussian Pell-Lucas dizilerini tanımladık. Gaussian Pell ve Gaussian Pell-Lucas dizilerinin Binet formüllerini ve üreteç fonksiyonlarını verdik. Üstelik, Gaussian Pell ve Gaussian Pell-Lucas sayılarını içeren bazı önemli özdeşlikler elde ettik. AMS Classification. 11B37, 11B39.1 * [email protected], 8 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well known Generalized Fibonacci sequence {푈푛}, 푈푛+1 = 푝푈푛 + 푞푈푛−1 , 푈0 = 0 푎푛푑 푈1 = 1, and generalized Lucas sequence {푉푛} are defined by 푉푛+1 = 푝푉푛 + 푞푉푛−1 , 푉0 = 2 푎푛푑 푉1 = 푝, where 푝 and 푞 are nonzero real numbers and 푛 ≥ 1. For 푝 = 푞 = 1, we have classical Fibonacci and Lucas sequences. For 푝 = 2, 푞 = 1, we have Pell and Pell- Lucas sequences. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Problems in Relating Various Tasks and Their Sample Solutions to Bloomâ
The Mathematics Enthusiast Volume 14 Number 1 Numbers 1, 2, & 3 Article 4 1-2017 Problems in relating various tasks and their sample solutions to Bloom’s taxonomy Torsten Lindstrom Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Lindstrom, Torsten (2017) "Problems in relating various tasks and their sample solutions to Bloom’s taxonomy," The Mathematics Enthusiast: Vol. 14 : No. 1 , Article 4. Available at: https://scholarworks.umt.edu/tme/vol14/iss1/4 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 14, nos. 1,2&3, p. 15 Problems in relating various tasks and their sample solutions to Bloom’s taxonomy Torsten Lindstr¨om Linnaeus University, SWEDEN ABSTRACT: In this paper we analyze sample solutions of a number of problems and relate them to their level as prescribed by Bloom’s taxonomy. We relate these solutions to a number of other frameworks, too. Our key message is that it remains insufficient to analyze written forms of these tasks. We emphasize careful observations of how different students approach a solution before finally assessing the level of tasks used. We take the arithmetic series as our starting point and point out that the objective of the discussion of the examples here in no way is to indicate an optimal way towards a solution. -
Adopting Number Sequences for Shielding Information
Manish Bansal et al, International Journal of Computer Science and Mobile Computing, Vol.3 Issue.11, November- 2014, pg. 660-667 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 3, Issue. 11, November 2014, pg.660 – 667 RESEARCH ARTICLE Adopting Number Sequences for Shielding Information Mrs. Sandhya Maitra, Mr. Manish Bansal, Ms. Preety Gupta Associate Professor - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India [email protected] , [email protected] , [email protected] Abstract:-The advancement of technology and global communication networks puts up the question of safety of conveyed data and saved data over these media. Cryptography is the most efficient and feasible mode to transfer security services and also Cryptography is becoming effective tool in numerous applications for information security. This paper studies the shielding of information with the help of cryptographic function and number sequences. The efficiency of the given method is examined, which assures upgraded cryptographic services in physical signal and it is also agile and clear for realization. 1. Introduction Cryptography is the branch of information security. The basic goal of cryptography is to make communication secure in the presence of third party (intruder). Encryption is the mechanism of transforming information into unreadable (cipher) form. This is accomplished for data integrity, forward secrecy, authentication and mainly for data confidentiality. -
On the Connections Between Pell Numbers and Fibonacci P-Numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 27, 2021, No. 1, 148–160 DOI: 10.7546/nntdm.2021.27.1.148-160 On the connections between Pell numbers and Fibonacci p-numbers Anthony G. Shannon1, Ozg¨ ur¨ Erdag˘2 and Om¨ ur¨ Deveci3 1 Warrane College, University of New South Wales Kensington, Australia e-mail: [email protected] 2 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] 3 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] Received: 24 April 2020 Revised: 4 January 2021 Accepted: 7 January 2021 Abstract: In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices. Keywords: Pell sequence, Fibonacci p-sequence, Matrix, Representation. 2010 Mathematics Subject Classification: 11K31, 11C20, 15A15. 1 Introduction The well-known Pell sequence fPng is defined by the following recurrence relation: Pn+2 = 2Pn+1 + Pn for n ≥ 0 in which P0 = 0 and P1 = 1. 148 There are many important generalizations of the Fibonacci sequence. The Fibonacci p-sequence [22, 23] is one of them: Fp (n) = Fp (n − 1) + Fp (n − p − 1) for p = 1; 2; 3;::: and n > p in which Fp (0) = 0, Fp (1) = ··· = Fp (p) = 1. -
Figurate Numbers
Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard tally marks is still seen in the symbols on dominoes or playing cards, and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate. Bear with me while we begin with a few elementary observations. Any number, n greater than 1, can be represented by a linear arrangement of n counters. The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements. The counters that make up a number m can alternatively be grouped in pairs instead of ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication). Numbers of these two forms are of course known as even and odd respectively. An even number is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate. Figure 1. Representation of numbers by rows of counters, and of even and odd numbers by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd numbers is known as a gnomon. These do not of course exhaust the possibilities. 1 2 3 4 5 6 7 8 9 n 2 4 6 8 10 12 14 2.n 1 3 5 7 9 11 13 15 2.n + 1 Triples, Quadruples and Other Forms Generalising the divison into even and odd numbers, the counters making up a number can of course also be grouped in threes or fours or indeed any nonzero number k. -
TRIANGULAR REPUNIT—THERE IS but 1 a Repunit Is Any Integer That
Czechoslovak Mathematical Journal, 60 (135) (2010), 1075–1077 TRIANGULAR REPUNIT—THERE IS BUT 1 John H. Jaroma, Ave Maria (Received June 27, 2009) Abstract. In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit. Keywords: repunit, triangular number MSC 2010 : 11A99 A repunit is any integer that can be written in decimal notation as a string of 1’s. Examples include 1, 11, 111, 1111, 11111,... Furthermore, if we denote rn to be the nth repunit then it follows that 10n − 1 (1) rn = . 9 Triangular numbers are those integers that can be represented by the number of dots evenly arranged in an equilateral triangle. More specifically, they are the numbers 1, 3, 6, 10, 15,... It follows that the nth triangular number, tn is the sum of the first n consecutive integers beginning with 1. In particular, for n > 1 n(n + 1) (2) tn = . 2 Now, the number 1 is both triangular and a repunit. It is the objective of this note to illustrate that 1 is the only such number. To this end, we state and prove the following lemma. It incorporates a result that [1] asserts had been known by the early Pythagoreans. Appearing in Platonic Questions, Plutarch states that eight times any triangular number plus one is a square. The result is actually necessary and sufficient. We demonstrate it as such here. 1075 Lemma. The integer n is triangular if and only if 8n +1 is a square. Proof. If n = tn, then by (2), n(n + 1) 8 +1=4n2 +4n +1=(2n + 1)2. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
PELL FACTORIANGULAR NUMBERS Florian Luca, Japhet Odjoumani
PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 105(119) (2019), 93–100 DOI: https://doi.org/10.2298/PIM1919093L PELL FACTORIANGULAR NUMBERS Florian Luca, Japhet Odjoumani, and Alain Togbé Abstract. We show that the only Pell numbers which are factoriangular are 2, 5 and 12. 1. Introduction Recall that the Pell numbers Pm m> are given by { } 1 αm βm P = − , for all m > 1, m α β − where α =1+ √2 and β =1 √2. The first few Pell numbers are − 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025,... Castillo [3], dubbed a number of the form Ftn = n!+ n(n + 1)/2 for n > 1 a factoriangular number. The first few factoriangular numbers are 2, 5, 12, 34, 135, 741, 5068, 40356, 362925,... Luca and Gómez-Ruiz [8], proved that the only Fibonacci factoriangular numbers are 2, 5 and 34. This settled a conjecture of Castillo from [3]. In this paper, we prove the following related result. Theorem 1.1. The only Pell numbers which are factoriangular are 2, 5 and 12. Our method is similar to the one from [8]. Assuming Pm = Ftn for positive integers m and n, we use a linear forms in p-adic logarithms to find some bounds on m and n. The resulting bounds are large so we run a calculation to reduce the bounds. This computation is highly nontrivial and relates on reducing the diophantine equation Pm = Ftn modulo the primes from a carefully selected finite set of suitable primes. 2010 Mathematics Subject Classification: Primary 11D59, 11D09, 11D7; Secondary 11A55. -
Problem Solving and Recreational Mathematics
Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2012 Chapters 1–44 August 1 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11 7/18 7/25 8/1 Friday 6/29 7/6 7/13 7/20 7/27 8/3 ii Contents 1 Digit problems 101 1.1 When can you cancel illegitimately and yet get the cor- rectanswer? .......................101 1.2 Repdigits.........................103 1.3 Sortednumberswithsortedsquares . 105 1.4 Sumsofsquaresofdigits . 108 2 Transferrable numbers 111 2.1 Right-transferrablenumbers . 111 2.2 Left-transferrableintegers . 113 3 Arithmetic problems 117 3.1 AnumbergameofLewisCarroll . 117 3.2 Reconstruction of multiplicationsand divisions . 120 3.2.1 Amultiplicationproblem. 120 3.2.2 Adivisionproblem . 121 4 Fibonacci numbers 201 4.1 TheFibonaccisequence . 201 4.2 SomerelationsofFibonaccinumbers . 204 4.3 Fibonaccinumbersandbinomialcoefficients . 205 5 Counting with Fibonacci numbers 207 5.1 Squaresanddominos . 207 5.2 Fatsubsetsof [n] .....................208 5.3 Anarrangementofpennies . 209 6 Fibonacci numbers 3 211 6.1 FactorizationofFibonaccinumbers . 211 iv CONTENTS 6.2 TheLucasnumbers . 214 6.3 Countingcircularpermutations . 215 7 Subtraction games 301 7.1 TheBachetgame ....................301 7.2 TheSprague-Grundysequence . 302 7.3 Subtraction of powers of 2 ................303 7.4 Subtractionofsquarenumbers . 304 7.5 Moredifficultgames. 305 8 The games of Euclid and Wythoff 307 8.1 ThegameofEuclid . 307 8.2 Wythoff’sgame .....................309 8.3 Beatty’sTheorem . 311 9 Extrapolation problems 313 9.1 Whatis f(n + 1) if f(k)=2k for k =0, 1, 2 ...,n? . 313 1 9.2 Whatis f(n + 1) if f(k)= k+1 for k =0, 1, 2 ...,n? . -
Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and Their Explicit Formulas by Ian Walker Ju
Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and their Explicit Formulas By Ian Walker June 2011 MST 501 Research Project In partial fulfillment for the Master’s of Teaching Mathematics (M.S.T.) Portland State University, Portland, Oregon John Pell (1611-1685) 2 John Pell (1611-1685) An “obscure” English Mathematician • Part of the 17th century intellectual history of England and of Continental Europe. • Pell was married with eight children, taught math at the Gymnasium in Amsterdam, and was Oliver Cromwell’s envoy to Switzerland. • Pell was well read in classical and contemporary mathematics. • Pell had correspondence with Descartes, Leibniz, Cavendish, Mersenne, Hartlib, Collins and others. • His main mathematical focus was on mathematical tables: tables of squares, sums of squares, primes and composites, constant differences, logarithms, antilogarithms, trigonometric functions, etc. 3 John Pell (1611-1685) An “obscure” English Mathematician • Many of Pell’s booklets of tables and other works do not list himself as the author. • Did not publish much mathematical work. Is more known for his activities, correspondence and contacts. • Only one of his tables was ever published (1672), which had tables of the first 10,000 square numbers. • His best known published work is, “An Introduction to Algebra”. It explains how to simplify and solve equations. • Pell is credited with the modern day division symbol and the double-angle tangent formula. • Pell is best known, only by name, for the Pell Sequence and the Pell Equation. 4 John Pell (1611-1685) An “obscure” English Mathematician • Division Symbol: • Double-Angle Tangent Formula: 2tan tan2 2 • Pell Sequence: 1 tan p 2p p p 1, p 2,n 2 n n1 n2 0 1 • Pell Equation: 2 2 x 2y 1 5 John Pell (1611-1685) An “obscure” English Mathematician • Both the Pell Sequence and the Pell Equation are erroneously named after him. -
Patterns in Figurate Sequences
Patterns in Figurate Sequences Concepts • Numerical patterns • Figurate numbers: triangular, square, pentagonal, hexagonal, heptagonal, octagonal, etc. • Closed form representation of a number sequence • Function notation and graphing • Discrete and continuous data Materials • Chips, two-color counters, or other manipulatives for modeling patterns • Student activity sheet “Patterns in Figurate Sequences” • TI-73 EXPLORER or TI-83 Plus/SE Introduction Mathematics has been described as the “science of patterns.” Patterns are everywhere and may appear as geometric patterns or numeric patterns or both. Figurate numbers are examples of patterns that are both geometric and numeric since they relate geometric shapes of polygons to numerical patterns. In this activity you will analyze, extend, and describe patterns involving figurate numbers and make connections between numeric and geometric representations of patterns. PTE: Algebra Page 1 © 2003 Teachers Teaching With Technology Patterns in Figurate Sequences Student Activity Sheet 1. Using chips or other manipulatives, analyze the following pattern and extend the pattern pictorially for two more terms. • • • • • • • • • • 2. Write the sequence of numbers that describes the quantity of dots above. 3. Describe this pattern in another way. 4. Extend and describe the following pattern with pictures, words, and numbers. • • • • • • • • • • • • • • 5. Analyze Table 1. Fill in each of the rows of the table. Table 1: Figurate Numbers Figurate 1st 2nd 3rd 4th 5th 6th 7th 8th nth Number Triangular 1 3 6 10 15 21 28 36 n(n+1)/2 Square 1 4 9 16 25 36 49 64 Pentagonal 1 5 12 22 35 51 70 Hexagonal 1 6 15 28 45 66 Heptagonal 1 7 18 34 55 Octagonal 1 8 21 40 Nonagonal 1 9 24 Decagonal 1 10 Undecagonal 1 ..