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Applications of Fibonacci Numbers

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Applications of Fibonacci Numbers Applications of Fibonacci Numbers Appl ications of Fibonacci Numbers Volume 9 Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications edited by Frederic T. Howard Wake Forest University, Winston-Salem, North Carolina, U.S.A. Springer-Science+Business Media, B.Y. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6545-2 ISBN 978-0-306-48517-6 (eBook) DOI 10.1007/978-0-306-48517-6 Cover figure by John C. Turner Printed on acid-free paper AII Rights Reserved © Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint ofthe hardcover Ist edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. TABLE OF CONTENTS A REPORT ON THE TENTH INTERNATIONAL CONFERENCE . vii AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS . ix FOREWORD . xxi THE ORGANIZING COMMITTEES. xxiii LIST OF CONTRIBUTORS TO THE CONFERENCE . xxv INTRODUCTION . XXVll FIBONACCI, VERN AND DAN . xxix UNIVERSAL BERNOULLI POLYNOMIALS AND P-ADIC CONGRUENCES A mold Adelberg . 1 A GENERALIZATION OF DURRMEYER-TYPE POLYNOMIALS AND THEIR APPROX­ IMATION PROPERTIES Octavian Agratini .................. 9 FIBINOMIAL IDENTITIES Arthur T. Benjamin, Jennifer J. Quinn and Jeremy A. Rouse 19 RECOUNTING BINOMIAL FIBONACCI IDENTITIES Arthur T. Benjamin and Jeremy A. Rouse ........ 25 THE FIBONACCI DIATOMIC ARRAY APPLIED TO FIBONACCI REPRESENTATIONS Marjorie Bicknell-Johnson. 29 ON PURPLE PARROTS, FIBONACCI NUMBERS, AND COLOR THEORY Marjorie Bicknell-Johnson. 39 FINDING FIBONACCI IN A FRACTAL Nathan C. Blecke, Kirsten Fleming and George William Grossman 43 POSITIVE INTEGERS (a 2 + b2 )/(ab + 1) ARE SQUARES Jens-P. Bode and Heiko Harborth ............. 63 ON THE FIBONACCI LENGTH OF POWERS OF DIHEDRAL GROUPS C.M. Campbell, P.P. Campbell, H. Doostie and E.F. Robertson 69 v VI TABLE OF CONTENTS SOME SUMS RELATED TO SUMS OF ORESME NUMBERS Charles K. Cook . 87 SOME THOUGHTS ON ROOK POLYNOMIALS ON SQUARE CHESSBOARDS Dan Fielder . 101 PYTHAGOREAN QUADRILATERALS Robert Hochberg and Glenn Hurlbert . 109 A GENERAL LACUNARY RECURRENCE FORMULA F. T. Howard . 121 ORDERING WORDS AND SETS OF NUMBERS: THE FIBONACCI CASE Clark Kimberling . 137 SOME BASIC PROPERTIES OF A TRIBONACCI LINE-SEQUENCE Jack Y. Lee . 145 A TYPE OF SEQUENCE CONSTRUCTED FROM FIBONACCI NUMBERS Aihua Li and Sindhu Unnithan. 159 CULLEN NUMBERS IN BINARY RECURRENT SEQUENCES Florian Luca and Pantelimon Stanici1 . 167 A GENERALIZATION OF EULER'S FORMULA AND ITS CONNECTION TO FI­ BONACCI NUMBERS Jonathan F. Mason and Richard H. Hudson . 177 EXTENSIONS OF GENERALIZED BINOMIAL COEFFICIENTS Richard L. Ollerton and Anthony G. Shannon . 187 SOME PARITY RESULTS REGARDING t-CORE PARTITIONS Neville Robbins and M. V. Subbamo . 201 GENERALIZED PELL NUMBERS AND POLYNOMIALS A. G. Shannon and A.F. Homdam .. 213 A FURTHER NOTE ON LUCASIAN NUMBERS Lawrence Somer . 225 SOME CONSTRUCTIONS AND THEOREMS IN GOLD POINT GEOMETRY John C. Turner . 235 SOME APPLICATIONS OF TRIANGLE TRANSFORMATIONS IN FIBONACCI GEOM­ ETRY John C. Turner .. 247 CRYPTOGRAPHY AND LUCAS SEQUENCE DISCRETE LOGARITHMS William A. Webb . 263 DIVISIBILITY OF AN F-L TYPE CONVOLUTION Michael Wiemann and Curtis Cooper . 267 GENERATING FUNCTIONS OF CONVOLUTION MATRICES Yongzhi (Peter) Yang . 289 F-L REPRESENTATION OF DIVISION OF POLYNOMIALS OVER A RING Chizhong Zhou and F. T. Howard 297 SUBJECT INDEX . 309 A REPORT ON THE TENTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS The Tenth International Conference on Fibonacci Numbers and Their Applications held at Northern Arizona University in Flagstaff, Arizona from June 24-28, 2002 found over 70 enthusiastic Fibonacci number lovers from Australia, Canada, England, Germany, Italy, Japan, Mexico, New Zealand, Poland, Romania, Scotland and the USA gathered together to hear over 50 excellent presentations. The gathering was attended by both old and new Fibonacci friends, but it was sadly noted that several regulars were unable to be with us this year. They were both warmly remembered and greatly missed. A special thanks goes to organizer Cal Long and all the folks at Northern Arizona University for their hospitality and generosity for hosting this outstanding conference. Monday through Wednesday morning found us savoring a variety of talks on things the­ oretical, operational and applicable of a Fibonacci and related nature, with members sharing ideas while renewing old friendships and forming new ones. Later on Wednesday the group was doubly treated. After the morning talks, we were en­ tertained by mathemagician Art Benjamin's most impressive presentation; displaying his skills and cleverness by mentally performing challenging mathematical manipulations and zapping out magic squares as if (yes) by magic. After graciously sharing some of the secrets of his wizardry with us, he dazzled one and all by mentally and accurately multiplying two five place numbers to terminate his mesmerizing performance. That afternoon we were bussed to our second wonder of the day: The Grand Canyon. Here we were able to spend several hours gazing at nature's wondrous spectacle. Oh to be a condor for an hour! In the evening a steak dinner was catered for us as we exchanged social and mathematical dialogue to the background of exquisite scenic wonder at the edge of the canyon. On the way back to campus we were able to witness a magnificent display of stars but an arm length away in the clear Arizona night sky. On Thursday and Friday it was back to many more interesting, informative presentations and during the breaks we were treated to Peter Anderson's marvelous computer display of the many photographs he took of association members and their families enjoying the Canyon. The closing banquet on Friday night terminated with a special tribute to Calvin T. Long for his distinguished career of 50 years as a teacher, mentor, and researcher, as well as valued VB VIII A REPORT ON ... friend, contributor and supporter of The Fibonacci Association. He was both praised and roasted by President Fred T. Howard and former editor Jerry Bergum. After much laughter and tears, Cal received a standing ovation from his proud and grateful group of his friends and colleagues. After over an hour of cordial good-byes everyone eventually drifted away, vowing that, Lord willing, we'll all meet again in Braunschweig, Germany in 2004. Charles K. Cook AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS Professor Arnold Adelberg (pp. 1-8) Mathematics Department Grinnell College Grinnell, IA 50112 USA Octavian Agratini (pp. 9-18) Faculty of Mathematics and Informatics "Babes-Bolyai" University str. Kogalniceanu, Nr. 1 3400 Cluj-Napoca ROMANIA Professor Peter G. Anderson School of Computer Science and Technology Rochester Institute of Technology 102 Lomb Memorial Drive Rochester, NY 14623-5608 USA Professor Arthur T. Benjamin (pp. 19-24; 25-28) Department of Mathematics Harvey Mudd College Claremont, CA 91771 USA Professor Kenneth S. Berenhaut Department of Mathematics Wake Forest University Winston-Salem, NC 27109 USA ix x AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS Gerald Bergum 1029 Circle Drive Brookings, SD 57006 USA Marjorie Bicknell-Johnson (pp. 29-38; 39-42) 665 Fairlane Avenue Santa Clara, CA 95051 USA Paul Bien 204 West Jefferson A venue Fairfield, IA 52556 USA Mr. Nathan BIecke (pp. 43-62) 4201 Menton Flint, MI 48507 USA Jens-P. Bode (pp. 63-67) Diskrete Mathematik Technische Universitiit Braunschweig D-38023 Braunschweig GERMANY Tom C. Brown Department of Mathematics and Statistics Simon Fraser University Burnaby BC, V5A 1S6 CANADA Paul Bruckman 3025 Shattuck Avenue, Apt. C. Berkeley, CA 94705 USA Barbara Mae Brunner P.O. Box 762 Sedona, AZ 86339 USA Dr. Colin M. Campbell (pp. 69-85) The Mathematical Institute University of St. Andrews St. Andrews KY 16 9SS Fife, SCOTLAND AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS Xl P.P. Campbell (pp. 69-85) The Mathematical Institute University of St. Andrews St. Andrews KY 16 9SS Fife, SCOTLAND Walter Carlip University of California, Santa Barbara 408 Harvard Street Vestal, NY 13850 USA Jeff Chamberlein Department of Mathematics and Computer Science Ithaca College Ithaca, NY 14850 USA Professor Hei-Chi Chan Mathematical Sciences Department University of illinois at Springfield P.O. Box 19243 . Springfield, IL 62794 USA Dr. Charles Cook (pp. 87-99) Mathematics Department University of South Carolina at Sumter Sumter, SC 29150-2498 USA Professor Curtis Cooper (pp. 267-287) Department of Mathematics and Computer Science Central Missouri State University Warrensburg, MO 64093-5045 USA Professor Karl Dilcher Department of Mathematics, Statistics and Computer Science Dalhousie University Halifax NS, B3H 3J5 CANADA Xll AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS H. Doostie (pp. 69-85) Mathematics Department University for Teacher Education 49 Mofateh A venue Tehran, 15614 IRAN Eric S. Egge Department of Mathematics Gettysburg College Gettysburg, PA 17325 USA Michele Elia Dipt. di Elettronica Politecnico di Torino C.so Duca degli Abruzzi 24 1-10129 Torino ITALY Larry Ericksen P.O. Box 172 Millville, NJ 08332-0172 USA Professor Emeritus Dan Fielder (pp. 101-108) School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332-0250 USA Dr. Kristen Flemming (pp. 43-62) Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859 USA C. T. Flynn 8021 Cobblestone Des Moines, IA 50322 USA George Grossman (pp. 43-62) Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859 USA AUTHORS, COAUTHORS AND OTHER CONFERENCE PARTICIPANTS Xlll Professor Helen Grundman Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010-2899 USA Ben Gum Grinnell College 1905 Sunset Street Grinnell, IA 50112 USA Professor Dr.
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