Applications of Fibonacci Numbers: Volume 4 Proceedings of ‘The Fourth International Conference on Fibonacci Numbers and Their Applications’, Wake Forest University, N.C., U.S.A., July 30–August 3, 1990 PDF
Preview Applications of Fibonacci Numbers: Volume 4 Proceedings of ‘The Fourth International Conference on Fibonacci Numbers and Their Applications’, Wake Forest University, N.C., U.S.A., July 30–August 3, 1990
Applications of Fibonacci Numbers Applications of Fibonacci Numbers Volume 4 Proceedings of 'The Fourth International Conference on Fibonacci Numbers alld Their Applications', Wake Forest University, N.C., U.S.A., July 30-August 3, 1990 edited by G. E. Bergum South Dakota State University, Brookings, South Dakota, U.S.A. A.N.Philippou Ministry of Education, Nicosia, Cyprus and A. F. Horadam University of New England, Armidale, New South Wales, Australia .... " Springer Science+Business Media, B.V. Ubrary of Congress Cataloging-in-Publication Data: Le 89-24547 ISBN 978-94-010-5590-1 ISBN 978-94-011-3586-3 (eBook) DOI 10.1007/978-94-011-3586-3 Cover figure by Or Sabine Jâger Printed an acid-tree paper AII Rights Reserved @ 1991 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 No part of the material protected by this copyright notice may ba reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS AREPORT ON THE FOURTH INTERNATIONAL CONFERENCE... vii LISTOF CONTRIBUTORS TO THIS PROCEEDINGS IX FOREWORD xv THE ORGANIZING COMMITTEES XVII LISTOF CONTRIBUTORS TO THECONFERENCE xix INTRODUCTION xxiii AFIBONACCI-BASED PSEUDO-RANDOM NUMBERGENERATOR Peter G. Anderson .1 ON THE PROOF OF GCD AND LCM EQUALITIES CONCERNING THE GENERALIZED BINOMIAL AND MULTINOMIAL COEFFICIENTS Shiro AndoandDaihachiro Sato 9 SUPERCUBE Joseph Arkin, DavidC. Arney, Lee S. DewaldandFrank R. Giordano 17 A NOTEON FUNDAMENTAL PROPERTIES OF RECURRING SERIES Joseph Arkin, David C. Arney, Frank R. Giordano and RickeyA. Kolb 33 PERIOD PATTERNS OF CERTAIN SECOND-ORDERLINEAR RECURRENCES MODULO APRIME David Banksand LawrenceSomer 37 NEARLY ISOSCELESTRIANGLES WHERETHE VERTEX ANGLE IS A MULTIPLE OF THE BASE ANGLE Marjorie Bicknell-Johnson 41 THE RING OF FIBONACCI (FIBONACCI "NUMBERS" WITH MATRIX SUBSCRIPT) Odoardo Brugia, Piero Filipponiand Francesco Mazzarella 51 ONE-RELATOR PRODUCTS OF CYCLIC GROUPS AND FIBONACCI-LIKE SEQUENCES C. M. Campbell, P. M. Heggie, E. F. Robertson and R. M. Thomas 63 AGENERALIZATION OFTHE FIBONACCI SEARCH Renato M. Capocelli 69 PASCAL'S TRIANGLE: TOP GUN ORJUST ONE OFTHE GANG? DanielC. Fielderand Cecil O. Alford 77 CONVERSION OF FIBONACCI IDENTITIES INTO HYPERBOLIC IDENTITIES VALID FORAN ARBITRARY ARGUMENT Piero Filipponiand Herta T. Freitag 91 DERIVATIVE SEQUENCES OF FIBONACCI AND LUCAS POLYNOMIALS Piero FilipponiandAlwyn F. Horadam 99 ACARRYTHEOREM FOR RATIONAL BINOMIAL COEFFICIENTS Dan Flath andRhodes Peele. . .... ...... ...... . 109 ON CO-RELATED SEQUENCES INVOLVING GENERALIZED FIBONACCI NUMBERS Herta T. Freitag and George M. Phillips 121 FIBONACCI AND B-ADICTREES IN MOSAIC GRAPHS Heiko Harborth andSabine Jager .127 FIBONACCI REPRESENTATIONSOF GRAPHS Heiko Harborth andArnfriedKemnitz .133 ON THE SIZES OF ELEMENTS IN THE COMPLEMENTOF ASUBMONOID OF INTEGERS Chung-wu Ho, James L. Parish, andJau-shyong Shiue 139 GENOCCHI POLYNOMIALS A. F. Horadam .145 AN APPLICATION OF ZECKENDORF'S THEOREM Roger V. Jean 167 v vi TABLEOF CONTENTS ANEW KIND OF GOLDEN TRIANGLE Clark Kimberling•................................................ .171 TERMS COMMON TO TWO SEQUENCES SATISFYING THE SAME LINEAR RECURRENCE Clark Kimberling .177 RECURRENCE RELATIONS IN EXPONENTIAL FUNCTIONS AND IN DAMPED SINUSOIDS AND THEIRAPPLICATIONS IN ELECTRONICS Joseph Lahr....................•................................189 SOME BASIC PROPERTIES OFTHE FIBONACCI LINE-SEQUENCE Jack Y. Lee ......•.•............................................203 DE MOIVRE-TYPE IDENTITIES FORTHETETRABONACCI NUMBERS Pin-Yen Lin 215 TWO GENERALIZATIONS OF GOULD'S STAROF DAVIDTHEOREM Calvin LongandShiro Ando 219 ON TRIANGULARLUCAS NUMBERS Ming Luo 231 AFAST ALGORITHM OFTHE CHINESE REMAINDERTHEOREM AND ITS APPLICATION TO FIBONACCI NUMBERS Kenji Nagasaka. Jau-Shyong Shiue and Chung-Wu Ho .241 GENERATING THE PYTHAGOREAN TRIPLES VIA SIMPLE CONTINUED FRACTIONS A. G. Schaake andJ. C. Turner 247 ON THE MOEBIUS KNOT TREEAND EUCLID'S ALGORITHM A. G. Schaake andJ. C. Turner 257 GENERALIZED FIBONACCI AND LUCAS FACTORIZATIONS A. G. Shannon, R. P. Loh andA. F. Horadam 271 ON EVEN FIBONACCI PSEUDOPRIMES Lawrence Somer............•.....................................277 POSSIBLE RESTRICTED PERIODS OF CERTAIN LUCAS SEQUENCES MODULO P LawrenceSomer 289 USING MATRIXTECHNIQUES TO ESTABLISH PROPERTIES OF AGENERALIZED TRIBONACCISEQUENCE Marcellus E. Waddill 299 SUBJECT INDEX. ... .... ...... ... .... .. .... ..... ... ... ... .. .... .... 309 AREPORT ON THE FOURTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS Sponsoredjointly by the Fibonacci Association and Wake Forest University, The Fourth International Conference on Fibonacci Numbers and Their Applications was held from July 30th to August 3rd, 1990. As the Conference took place at Wake Forest University, our foreign visitors especially gained a most enjoyable insight into one of America's delightful set-ups: a small, highly esteemed, liberal arts University, nestled at the outskirts of a faithfully restored 18thcentury town - Winston-Salem, N.C. Immediately upon arrival it became clear to us how carefully and competently -under the leadership of the co-chairmen of the International Committee, A. F. Horadam (Australia) and A. N. Philippou (Cyprus), as well as of the co-chairmen of the Local Committee, F. T. Howard and M. E. Waddill -our Conference had been planned and prepared. Special thanks must also go to G.E. Bergum, editor of our Fibonacci Quarterly Journal, for arranging an outstanding program. There were about 50 participants, 40 of them presented papers, of these, two were women. From some 13 different lands they came, beside the U.S., the host country, Italy would have won the prize for maximum attendance, then Canada and Scotland, closely followed by Australiaand Japan. Papers related to the Fibonacci numbers and their ramifications, and to recursive sequences and their generalizations, as well as those which analyzed and explained number relationships, were presented. Once again, as had been the case in our previous conference, did the diversity of the papers give testimony to the fertility of Fibonacci-related mathematics, as well as to the fructification of ideas, brought about through our mutual, but at the same time, disparate interests. The interplay between theoretically oriented manuscripts and those which highlighted practical aspects, wasagain conspicuous and fascinating. The Conference was held in the new Olin Physical Laboratory which was accessible via overcoming several road hurdles which had become necessitated by construction work across the campus. Although our hosts were most apologetic about this, we saw it as a sign of a vital, dynamicand indeed, growing University. Once in our medium-sized auditorium, we wereintrigued (and assisted) by "the wonders technology had wrought": there were two overhead projectors and blackboards - ugh, whiteboards (!) - came from everywhere; up and down they went, above and below, over and across, sometimes interceded by a screen which appeared from nowhere...,and all ofit happened by the touch ofa button, skillfully activated by the cognoscenti. Ofcourse, there was not only food for the mind and the soul, but also for the stomach. Wake Forest University graciously treated us to daily morning and afternoon coffee breaks, and the president, Dr. Thomas K. Hearn, Jr., hosted a wineand cheese reception on campus. Even though our daily meetings took place from 9:00 a.m. till noon, and from 2:00 p.m. to 5:00 p.m., we did not ALWAYS work. In midweek, the afternoon was freed, and we took off to Doughton Park in the beautiful Blue Ridge Mountains of North Carolina. There the group dispersed to enjoy the magnificent scenery with a choice of several hiking trails that offer spectacular vistas. Those of us who preferred less energetic activities, relaxed at a coffee shop where we did, what weseem to bedoing best, orat least, most often, and with pleasure: vii viii AREPORT ON ... exchange mathematical ideas. All this was followed by a lavish, typically North Carolinian dinner at Shatley Springs. The next day we celebrated our customary evening banquet. It was held on campus, and was at onceelegantand friendly, somehow reflecting thespirit ofourgroup. Wespeak with many different foreign accents. And yet, we all understand each other, professionally, and personally. The magnetism ofour beloved discipline hassomehow promoted a very special bond offriendship. Manyofus had been together in someofthe past conferences. Quitea few papers exhibited the resulting kindling of common mathematical interests which culminated in joint authorships. Maybe, several of you are already gathering your thoughts for our next Conference. "AufWiedersehen", then, in 1992at St. Andrews University, Scotland. Herta T. Freitag LIST OF CONTRIBUTORSTO THIS PROCEEDINGS PROFESSORCECIL O. ALFORD (pp. 77-90) SCHOOL OF ELECTRICAL ENGINEERING GEORGIA INSTITUTE OFTECHNOLOGY ATLANTA, GEORGIA 30332-0250 PROFESSORPETERG. ANDERSON (pp. 1-8) SCHOOL OF COMPUTERSCIENCEAND TECHNOLOGY ROCHESTERINSTITUTE OF TECHNOLOGY ONE LOMB MEMORIAL DRIVE POSTOFFICE BOX 9887 ROCHESTER, NY 14623-0887 PROFESSORSHIRO ANDO (pp. 9-16; 219-230) COLLEGE OF ENGINEERING HOSEI UNIVERSITY 3-7-2, KAJINO-CHO KOGANEI-SHI TOKYO 184, JAPAN MR. JOSEPH ARKIN (pp. 17-32; 33-36) 197OLD NYACKTURNPIKE SPRING VALLEY, NY 10977 LT. COL. DAVID C. ARNEY (pp. 17-32; 33-36) DEPARTMENT OF MATHEMATICS UNITED STATES MILITARY ACADEMY WEST POINT, NY 10996-1786 MR. DAVID BANKS (pp. 37-40) 213 VALLEY PARK DRIVE CHAPEL HILL, NC 27514 DR. MARJORIE BICKNELL-JOHNSON (pp. 41-50) 665 FAIRLANE AVENUE SANTA CLARA, CA 95051 PROFESSORTHOMAS C. BROWN DEPARTMENTOF MATHEMATICS AND STATISTICS SIMON FRASERUNIVERSITY BURNABY, BRITISH COLUMBIA CANADA V5A IS6 .This listalso includesthoseauthors whose paperswere published elsewhereor were not accepted. ix x CONTRIBUTORS TOTHIS PROCEEDINGS MR. ODOARDO BRUGIA (pp. 51-62) FONDAZIONE UGO BORDONI VIA B. CASTIGLIONE, 59 1-00142 ROMA, ITALY DR. COLIN M. CAMPBELL (pp. 63-68) UNIVERSITYOF ST. ANDREWS DEPARTMENT OF MATHEMATICS AND COMPUTERSCIENCE MATHEMATICAL INSTITUTE, NORTH HAUGH ST. ANDREWS KY16 9SS FIFE, SCOTLAND PROFESSOR RENATO M. CAPOCELLI (pp. 69-76) DIPARTIMENTO DI MATEMATICA UNIVERSITA' DI ROMA "LA SAPIENZA" 00185 ROMA ITALY COL. LEE S. DEWALD (pp. 17-32) DEPARTMENTOF MATHEMATICS UNITED STATES MILITARY ACADEMY WEST POINT, NY 10996-1786 PROFESSOR DANIELC. FIELDER (pp. 77-90) SCHOOL OF ELECTRICAL ENGINEERING GEORGIA INSTITUTEOF TECHNOLOGY ATLANTA, GEORGIA 30332-0250 MR. PIERO FILIPPONI (pp. 51-62; 91-98; 99-108) FONDAZIONE UGO BORDONI VIA B. CASTIGLIONE, 59 1-00142 ROMA, ITALY PROFESSOR DANIEL FLATH (pp. 109-120) DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTYCOURTSOUTH NUMBERTHREE UNIVERSITY OF SOUTH ALABAMA MOBILE, ALABAMA 36688 PROFESSOR HERTA T. FREITAG (pp. 91-98; 121-125) B-40 FRIENDSHIP MANOR 320 HERSHBERGERROAD, N.W. ROANOKE, VA 24012 LT. COL. FRANK R. GIORDANO (pp. 17-32; 33-36) DEPARTMENTOF MATHEMATICS UNITED STATES MILITARY ACADEMY WEST POINT, NY 10996-1786 CONTRIBUTORS TO THIS PROCEEDINGS Xl PROFESSOR DR. HEIKO HARBORTH (pp. 127-132; 133-138) BIENRODERWEG 47 D-3300 BRAUNSCHWEIG WEST GERMANY PROF. P. M. HEGGIE (pp. 63-68) UNIVERSITYOF ST. ANDREWS DEPARTMENT OF MATHEMATICS AND COMPUTERSCIENCE MATHEMATICAL INSTITUTE, NORTH HAUGH ST. ANDREWS KY16 9SS FIFE, SCOTLAND PROFESSORCHUNG-WU HO (pp. 139-144; 241-246) DEPARTMENTOF MATHEMATICS AND STATISTICS BOX 1653 SOUTHERN ILLINOIS UNIVERSITY AT EDWARDSVILLE EDWARDSVILLE, IL 62026-1653 PROFESSORA. F. HORADAM (pp. 99-108; 145-166; 271-276) DEPARTMENTOF MATHEMATICS, STATISTICS AND COMPUTERSCIENCE THE UNIVERSITYOF NEW ENGLAND ARMIDALE, NEW SOUTH WALES 2351 AUSTRALIA DR. SABINE JAGER (pp. 127-132) BULTENWEG 7 D-3300 BRAUNSCHWEIG WESTGERMANY PROFESSOR ROGER V. JEAN (pp. 167-170) DEPARTMENTOF MATHEMATICS AND COMPUTERSCIENCE A UNIVERSITE DU QUEBEC RIMOUSKI 300, ALLEE DES URSULINES, REMOUSKI QUEBEC, G5L 3A1 CANADA PROFESSORARNFRIED KEMNITZ (pp. 133-138) WUMMEWEG 10 3300 BRAUNSCHWEIG WEST GERMANY PROFESSORCLARK KIMBERLING (pp. 171-176; 177-188) DEPARTMENT OF MATHEMATICS UNIVERSITYOF EVANSVILLE 1800 LINCOLN AVENUE EVANSVILLE, INDIANA 47722 LT. COL. RICKEY A. KOLB (pp. 33-36) DEPARTMENT OF MATHEMATICS UNITED STATES MILITARY ACADEMY WEST POINT, NY 10996-1786
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