Table Of ContentApplications 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
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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
