Applications of Fibonacci Numbers

Appl ications of Fibonacci Numbers

Volume 5

Proceedings of 'The Fifth International Conference on Fibonacci Numbers and Their Applications',

The University of St. Andrews, Scotland, July 20-July 24, 1992

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.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-4912-2 ISBN 978-94-011-2058-6 (eBook) DOI 10.1007/978-94-011-2058-6

Covar figura by Haiko Harborth

Printed on acid-free paper

AII Rights Reserved © 1993 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1 st edition 1993

No part of the material protected by this copyright notice may be 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

A REPORT ON THE FIFTH INTERNATIONAL CONFERENCE ixLIST OF CONTRIBUTORS TO THIS PROCEEDINGS xiFOREWORD xxvTHE ORGANIZING COMMITTEES xxviiLIST OF CONTRIBUTORS TO THE CONFERENCE xxixINTRODUCTION xxxiii

MULTIDIMENSIONAL GOLDEN MEANSPeter G. Anderson 1

A NECESSARY AND SUFFICIENT CONDITION THAT RAYS OF A STARCONFIGURATION ON PASCAL'S TRIANGLE COVER ITS CENTER WITH RESPECT TOGCD AND LCM

Shiro Ando and Daihachiro Sato 11ON THE MINIMAL CENTER COVERING STARS WITH RESPECT TO GCD IN PASCAL'SPYRAMID AND ITS GENERALIZATIONS

Shiro Ando and Daihachiro Sato 37AN EXTENSION OF AN OLD CLASSICAL DIOPHANTINE PROBLEM

Joseph Arkin, David C. Arney, Frank R. Giordano, Rickey A. Kolband Gerald E. Bergum 45

FOURIER ANALYSIS IN FINITE NILPOTENT GROUPSHuseyin Aydin and Geoff C. Smith 49

WALL AND VINSON REVISITEDHuseyin Aydin, Ramazan Dikici and Geoff C. Smith 61

DESIGNER DECIMALS: FRACTIONS WHICH CONTAIN SECOND ORDER RECURSIONSEQUENCES IN THEIR DECIMAL EXPANSIONS, READING LEFT TO RIGHT ORRIGHT TO LEFT

Marjorie Bicknell-Johnson 69SEMIGROUP PRESENTATIONS AND NUMBER SEQUENCES

Colin M. Campbell, Edmund F. Robertson and Richard M. Thomas 77COUNTING THE NUMBER OF SOLUTIONS OF CONGRUENCES

Umberto Cerruti 85ON THE NUMBER OF OVERLAPPING SUCCESS RUNS IN A SEQUENCE OFINDEPENDENT BERNOULLI TRIALS

O. Chryssaphinou, S. Papastavridis and T. Tsapelas 103EMBEDDING FIBONACCI WORDS INTO FIBONACCI WORD PATTERNS

Wai-fong Chuan 113ON SUMS OF CUBES OF FIBONACCI NUMBERS

Stuart Clary and Paul D. Hemenway 123INTEGER SIDED TRIANGLES WHOSE RATIO OF ALTITUDE TO BASE IS ANINTEGER

Charles K. Cook and Gerald E. Bergum 137ON A CLASS OF ITERATIVE RECURRENCE RELATIONS

Karl Dilcher 143DISCRETE OPERATIONAL CALCULI FOR TWO-SIDED SEQUENCES

Ivan H. Dimovski and Virginia S. Kiryakova 159TRANSFORMATIONS OF FIBONACCI-LUCAS IDENTITIES

L. A. G. Dresel 169FIBONACCI NUMBERS AND CONTINUED FRACTION EXPANSIONS

Michael Drmota 185

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vi TABLE OF CONTENTS

THE DFF AND DFFz TRIANGLES AND THEIR MATHEMATICAL PROPERTIESMarco Faccio, Giuseppe Ferri and Arnaldo D'Amico 199

CONTRIBUTIONS FROM CASCADED COMBINATIONS TO THE NAMING OF SPECIALPERMUTATIONS

Daniel C. Fielder and Cecil O. Alford 207THE ZECKENDORF REPRESENTATION OF {Fkn/Fn}

Piero Filipponi and Herta T. Freitag 217FRACTAL PATTERNS DERIVED FROM RATIONAL BINOMIAL COEFFICIENTS

Dan Flath and Rhodes Peele 221HAUSDORFF DIMENSION IN PASCAL'S TRIANGLE

Dan Flath and Rhodes Peele 229CHAOS, ELLIPTIC CURVES AND ALL THAT

Matthew Fletcher and Geoff C. Smith 245CO-RELATED SEQUENCES SATISFYING THE GENERAL SECOND ORDERRECURRENCE RELATION

H. T. Freitag and G. M. Phillips 257ON THE MOMENTS OF THE SUM-OF-DIGITS FUNCTION

P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy 263PROBLEMS ON FIBONACCI AND B-ADIC TREE REPRESENTATIONS BY REGULAR K­GONS

Heiko Harborth 273TWO-DISTANCE SETS AND THE GOLDEN RATIO

Heiko Harborth and Lothar Piepmeyer 279A FIBIONACCI PROBLEM CLASSIFICATION SCHEME USEFUL TO UNDERGRADUATEPEDAGOGY

Russell Jay Hendel 289ZECKENDORF REPRESENTATIONS OF POSITIVE AND NEGATIVE INTEGERS BYPELL NUMBERS

A. F. Horadam 305INTEGRATION SEQUENCES OF FIBONACCI AND LUCAS POLYNOMIALS

Alwyn F. Horadam and Piero Filipponi 317PELL-TYPE NUMBER GENERATORS OF PYTHAGOREAN TRIPLES

A. F. Horadam and A. G. Shannon 331BALANCE MORPHOLOGY OF A BINARY TREE

Yasuichi Horibe 345NORLUND'S NUMBER B(n)

nF. T. Howard 355

ON STIRLING-TYPE PAIRS AND EXTENDED GEGENBAUER-HUMBERT-FIBONACCIPOLYNOMIALS

Leetsch C. Hsu 367A SEQUENCE ARISING FROM REFLECTIONS IN MULTIPLE GLASS PLATES

Naotaka Imada 379DIOPHANTINE REPRESENTATION OF NON-FIBONACCI NUMBERS

James P. Jones 387ON SOME PROBLEMS RELATED TO THE FIBONACCI GROUP

Ann Chi Kim 393CONGRUENCE PROPERTIES OF FIBONACCI NUMBERS AND FIBONACCICOEFFICIENTS MODULO p 2

William A. Kimball and William A. Webb 399ORDERINGS OF THE SET OF ALL POSITIVE FIBONACCI SEQUENCES

Clark Kimberling 405

TABLE OF CONTENTS VII

SOME RESULTS CONCERNING THE RECIPROCAL SUM OF PRIME DIVISORS OF ALUCAS NUMBER

Peter Kiss .417RATIONAL NUMBERS WITH PREDICTABLE ENGEL PRODUCT EXPANSIONS

Arnold Knopfmacher .421TWO FAMILIES OF ORTHOGONAL POLYNOMIAL SYSTEMS RELATED TOFIBONACCI CHAINS

Wolfdieter Lang .429ON THE FORMATION OF HIGHER ORDER HIGHER DIMENSIONAL LINE-SEQUENTIALVECTOR SPACES

Jack Y. Lee 441ARITHMETIC SEQUENCES AND SECOND ORDER RECURRENCES

Calvin Long, Graeme L. Cohen, Timothy Langtry and A. G. Shannon .449GENERALIZED FIBONACCI PSEUDOPRIMES AND PROBABLE PRIMES

Winfried B. Muller and Alan Oswald .459A GENERALIZATION OF THE FIBONACCI SEQUENCE

S. P. Pethe and C. N. Phadte .465HOW TO ADVANCE ON A STAIRWAY BY COIN FLIPPINGS

Helmut Prodinger 473FIBONACCI SEQUENCE AND ITS GENERALIZATIONS HIDDEN IN ALGORITHMS FORGENERATING MORSE CODES

Shuichi Sato 481ON MATRIX REPRESENTATIONS OF GENERALIZED FIBONACCI NUMBERS ANDTHEIR APPLICATIONS

Shuichi Sato 487A CHOLESKY DECOMPOSITION IN MATCHING INSULIN PROFILES

A. G. Shannon, R. L. Ollerton and D. R. Owens .497GENERALIZED EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS

John C. Sjoberg 507DIVISIBILITY OF TERMS IN LUCAS SEQUENCES BY THEIR SUBSCRIPTS

Lawrence Somer 515UPPER BOUNDS FOR FREQUENCIES OF ELEMENTS IN SECOND-ORDERRECURRENCES OVER A FINITE FIELD

Lawrence Somer 527ON THE CONVERGENCE OF QUOTIENTS OF SOME RECURSIVE SEQUENCES

Lea Terracini 547TWO DISTRIBUTION PROBLEMS FOR POLYNOMIALS

Robert F. Tichy 561THE ELEMENTS OF ENTEGER GEOMETRY

J. C. Turner and A. G. Schaake 569TOTIENT FUNCTIONS ON THE EULER NUMBER TREE

J. C. Turner, H. Garcia and A. G. Schaake 585USING MATRIX TECHNIQUES TO ESTABLISH PROPERTIES OF k-ORDER LINEARRECURSIVE SEQUENCES

Marcellus E. Waddill 601SUBJECT INDEX 617

A REPORT ONTHE FIFTH INTERNATIONAL CONFERENCE

ONFIBONACCI NUMBERS AND THEIR APPLICATIONS

Ever since our previous Meeting at Wake Forest University in North Carolina, the 1992Conference had been awaited with keen anticipation. Finally, the announcement appeared:"sponsored jointly by The Fibonacci Association and The University of St. Andrews. THEFIFTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIRAPPLICATIONS will be held at The University of St. Andrews, Scotland, from July 20th toJuly 24th 1992. Co-chairmen of the Local Committee are George M. Phillips and Colin M.Campbell, whereas the International Committed is co-chaired by A.N. Philippou and A.F.Horadam."

The participation, 80 in number, 12 of whom are women mathematicians, practicallydoubled previous attendances. All five continents were represented. From Europe there were36; 29 came from America, 10 from Asia, 4 from Australia, and 1 from Africa. Among the 24countries represented by Conference participants, the United States provided the largestcontingent of 25 followed by Scotland and England, each with 8, and four countries-Austria,Canada, Italy, and Japan-each providing four registrants.

In all our Conferences do we greatly appreciate A.N. Philippou, "FATHER OF OURINTERNATIONAL CONFERENCES," as he has initiated our FIRST meeting at PatrasUniversity in Greece in 1984. And in all our Conferences (and I do hope that in his proverbialmodesty he will not censure this remark) we always cherish our conviction that a program,designed by our esteemed and beloved editor, Professor G.E. Bergum, spells excellence, even if-alas-this time double sessions would become necessary.

What caused the big increase in attendance?

It may have been the fact that The University of St. Andrews is held in high esteem theworld over. It may have been the magnetism, mathematical as well as personal, of the set of co­chairmen.

Soul-searching choice decisions had to be made for the overlapping sessions as there were70 papers, 6 of them presented by women mathematicians who hailed from Bulgaria, China,Italy, Scotland, and (two of them) from the U.S. At least three "non-mathematicians" gavepapers, one a research astronomer, two electrical engineers. The ages ranged from 33- to 83+,an age span of 50 years! And the distance traveled by speakers ranged from zero (four St.Andrews facuIty members gave papers) to approximately 12,000 miles (the journey from NewZealand).

Our home during the Conference was a University dormitory. John Burnet Hall,formerly a hotel, and still providing the comfort of such. Colleges and Universities have thereputation of offering dull, institutionalized fare. Our food, taken at the dorm's cafeteria,constituted an enjoyable counterexample.

St. Andrews is an ancient institution. And during its nearly six centuries of existence, ithas maintained vigorous scholarly impact across the whole academic spectrum. St. Andrews hasbeen called "a gem of a University"-uniquely Scottish by history and beautiful location, yetunusually cosmopolitan.

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x A REPORT ON...

The Conference's social events rounded off, and enhanced, our academic sessions. Thetraditional mid-conference's afternoon excursion took us to Falkland, a Renaissance Palace,which grew out of the medieval Falkland Castle. At once did we get lured into the quaintnessof an historically rich palace and became enchanted by the charming multi-coloredness of thegarden.

To convey, the congenial and happy atmosphere at our Conference-dinner adequatelywould require a vocabulary far richer than mine. Interspersed with inspirational short talks andremarks, animated by delicious banquet fare and, most of all, by having our whole groupgathered together, it was simply delightful.

And, finally, the Conference itself.

Erudite and always carefully prepared papers ranged over the heights and depths of"purity" and "applicability," once more illustrating the startling way in which these two facetsof mathematics are duals of each other. And while we speak with many different accents, weunderstand each other on a much more significant level. Almost immediately, friendshipsblossomed or ripened, as the love of our discipline and the enthusiasm for it were written overall the faces of the "Fibonaccians" as some of us like to refer to ourselves. That one week inScotland, kindled by the serenity of the Scottish landscape and enhanced by the spirit of ourScottish hosts and co-mathematicians, gave us experiences which were both mentally enrichingand personally heartwarming.

Finally, it was "farewell." But it is with much happiness that we can say: "AufWiedersehen in two years at Pullman, Washington."

Herta T. Freitag

LIST OF CONTRIBUTORS TO THIS PROCEEDINGS·

PROFESSOR CECIL O. ALFORD (pp. 207-216)SCHOOL OF ELECTRICAL ENGINEERINGGEORGIA INSTITUTE OF TECHNOLOGYATLANTA, GEORGIA 30332-0250

PROFESSOR PETER G. ANDERSON (pp. 1-10)SCHOOL OF COMPUTER SCIENCE AND TECHNOLOGYROCHESTER INSTITUTE OF TECHNOLOGYONE LOMB MEMORIAL DRIVEPOST OFFICE BOX 9887ROCHESTER, NY 14623-0887

PROFESSOR SHIRO ANDO (pp. 11-36; 37-44)COLLEGE OF ENGINEERINGHOSEI UNIVERSITY3-7-2, KAJINO-CHOKOGANEI-SHI, TOKYO 184JAPAN

MR. JOSEPH ARKIN (pp. 45-48)197 OLD NYACK TURNPIKESPRING VALLEY, NY 10977

COL. DAVID C. ARNEY (pp. 45-48)DEPARTMENT OF MATHEMATICSUNITED STATES MILITARY ACADEMYWEST POINT, NY 10996-1786

DR. HUSEYIN AYDIN (pp. 49-60; 61-68)ATATURK UNIVERSITYFEN-ED FACULTYDEPARTMENT OF MATHEMATICSERZURUM,TURKEY

DR. MICHEL BALLIEU29 B 5 PLACE MATTEOTTIB 1700 LA LOUVIEREBELGIUM

.This list includes all authors and coauthors of papers presented at the conference even if theirpaper was rejected, published elsewhere or not submitted to the proceedings. Those whoattended but did not present a paper are also in this list.

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PROFESSOR GERALD E. BERGUM (pp. 45-48; 137-142)COMPUTER SCIENCE DEPARTMENTSOUTH DAKOTA STATE UNIVERSITYBOX 2201BROOKINGS, SD 57007-0194

DR. MARJORIE BICKNELL-JOHNSON (pp. 69-76)665 FAIRLANE AVENUESANTA CLARA, CA 95051

PROFESSOR NGUYEN-HUU BONGDEPARTMENT OF MATHEMATICSUNIVERSITI BRUNEI DARUSSALAMGADONG 3186 BRUNEI

DR. RUTH BOWER525-55TH STREETWEST PALM BEACH, FL 33407-2607

PROFESSOR A. V. BOYDDEPARTMENT OF STATISTICS & ACTUARIAL SCIENCEUNIVERSITY OF THE WITWATERSRAND1 JAN SMUTS AVENUEJOHANNESBURG 2050, SOUTH AFRICA

DR. COLIN M. CAMPBELL (pp. 77-84)UNIVERSITY OF ST ANDREWSDEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCEMATHEMATICS INSTITUTE, NORTH HAUGHST ANDREWS KY16 9SSFIFE, SCOTLAND

PROFESSOR UMBERTO CERRUTI (pp. 85-102)UNIVERSITA DI TORINODIPARTIMENTO DI MATEMATICAVIA PRINCIPE AMEDEO 810123 TORINO, ITALIA

DR. OURANIA CHRYSSAPHINOU (pp. 103-112)UNIVERSITY OF ATHENSDEPARTMENT OF MATHEMATICSSTATISTICAL UNITPANEPISTEMIOPOLIS, 15784ATHENS, GREECE

PROFESSOR WAI-FONG CHUAN (pp. 113-122)DEPARTMENT OF MATHEMATICSCHUNG-YUAN CHRISTIAN UNIVERSITYCHUNG LI, TAIWAN 32023REPUBLIC OF CHINA

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR JEN-CHUNG CHUANDEPARTMENT OF MATHEMATICSNATIONAL TSING HUA UNIVERSITYHSINCHU, TAIWAN 300REPUBLIC OF CHINA

DR. STUART CLARY (pp. 123-136)2233 TYRONE STREETAKRON, OHIO 44312

PROFESSOR GRAEME 1. COHEN (pp. 449-458)SCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF TECHNOLOGY, SYDNEYP.O. BOX 123BROADWAY N. S. W. 2007AUSTRALIA

DR. CHARLES K. COOK (pp. 137-142)MATHEMATICS DEPARTMENTUNIVERSITY OF SOUTH CAROLINAAT SUMTERSUMTER, SC 29150-2498

PROFESSOR ARNALDO D'AMICO (pp. 199-206)DIPT. DI INGEGNERIA ELETTRONICAUNIVERSITA' DI TOR VERGATAVIA ORAZIO. RAIMONDO 1100173 ROMA, ITALIA

MR. RAMAZAN DIKICI (pp. 61-68)SCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF BATHCLAVERTON DOWN, BATH BA2 7AYENGLAND

PROFESSOR KARL DILCHER (pp. 143-158)DEPARTMENT OF MATHEMATICS, STATISTICS & COMPUTER SCIENCEDALHOUSE UNIVERSTIYHALIFAX, NOVA SCOTIACANADA B3H 3J5

PROFESSOR DR. IVAN H. DIMOVSKI (pp. 159-168)BULGARIAN ACADEMY OF SCIENCEINSTITUTE OF MATHEMATICSP.O. BOX 3731090 SOFIA, BULGARIA

DR. 1. A. G. DRESEL (pp. 169-184)3 WESTCOTE ROADREADING, BERKS RG3 2DLENGLAND

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DR. MICHAEL DRMOTA (pp. 185-198)INSTITUTE FUR ALGEBRA UNDDISCRETE MATHEMATICSTECHNISCHE UNIVERSITAT WIENWIEDNER HAUPTSTRASSE 8-10A-1040 WIRN, AUSTRIA

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR MARCO FACCIO (pp. 199-206)DIPT. DI INGEGNERIA ELETTRICAUNIVERSITA' DI L'AQUILALOCALITA' MONTELUCO DI ROIO67040 POGGIO DI ROIOL'AQUILA, ITALIA

PROFESSOR GIUSEPPE FERRI (pp. 199-206)DIPT. DI INGEGNERIA ELETTRICAUNIVERSITA' DI L'AQUILALOCALITA' MONTELUCO DI ROIO67040 POGGIO DI ROIOL'AQUILA,ITALIA

PROFESSOR DANIEL C. FIELDER (pp. 207-216)SCHOOL OF ELECTRICAL ENGINEERINGGEORGIA INSTITUTE OF TECHNOLOGYATLANTA, GEORGIA 30332-0250

MR. PIERO FILIPPONI (pp. 217-220; 317-330)FONDAZIONE UGO BORDONIVIALE BALDASSARRE CASTIGLIONE, 59I-00142-ROMA, ITALY

PROFESSOR DANIEL FLATH (pp. 221-228; 229-244)DEPARTMENT OF MATHEMATICS AND STATISTICSFACULTY COURT SOUTH NUMBER THREEUNIVERSITY OF SOUTH ALABAMAMOBILE, ALABAMA 36688

MR. MATTHEW FLETCHER (pp. 245-256)C/O DR. GEOFF C. SMITHSCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF BATHCLAVERTON DOWN, BATH BA2 7AYENGLAND

DR. DOROTHY M.E. FOSTERTHE MATHEMATICAL INSTITUTEUNIVERSITY OF ST. ANDREWSTHE NORTH HAUGHST. ANDREWS KY16 9SSFIFE, SCOTLAND

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR HERTA T. FREITAG (pp. 217-220; 257-262)B-40 FRIENDSHIP MANOR320 HERSHBERGER ROAD, N. W.ROANOKE, VA 24012-1927

DR. H. GARCIA (pp. 585-600)DEPARTMENT OF MATHEMATICSWAIARIKI POLYTECHNICROTORUA, NEW ZEALAND

COL. FRANK R. GIORDANO (pp. 45-48)DEPARTMENT OF MATHEMATICSUNITED STATES MILITARY ACADEMYWEST POINT, NY 10996-1786

PROFESSOR PETER GRABNER (pp. 263-272)INSTITUT FUR ALGEGRAUND DISKRETE MATHEMATIKTECHNISCHE UNIVERSITAT WIENWIEDNER HAUPTSTRASSE 8-10A-1040 WIEN, AUSTRIA

PROFESSOR DR. HEIKO HARBORTH (pp. 273-278; 279-288)BIENRODER WEG 47D-3300 BRAUNSCHWEIGWEST GERMANY

MS. DEBORAH H. HARRELL2335 SALEM COURTWINSTON-SALEM, NC 27108

DR. PAUL D. HEMENWAY (pp. 123-136)2808 ROLLINGWOOD DRIVEAUSTIN, TEXAS 78746

DR. RUSSELL JAY HENDEL (pp. 289-304)DEPARTMENT OF MATHEMATICSMORRIS COLLEGE100 WEST COLLEGE STREETSUMTER, SC 29150

PROFESSOR ALWYN F. HORADAM (pp. 305-316; 317-330; 331-344)DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCETHE UNIVERSITY OF NEW ENGLANDARMIDALE, N.S.W. 2351AUSTRALIA

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PROFESSOR YASUICHI HORIBE (pp. 345-354)DEPARTMENT OF COMPUTER SCIENCEFACULTY OF ENGINEERINGSHIZUOKA UNIVERSITYHAMAMATSU, 432 JAPAN

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR FRED T. HOWARD (pp. 355-366)DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCEBOX 7388, REYNOLDA STATIONWAKE FOREST UNIVERSITYWINSTON-SALEM, NC 27109

PROFESSOR LEETSCH C. HSU (pp. 367-378)DEPARTMENT OF MATHEMATICSDALIAN INSTITUTE OF TECHNOLOGYDALIAN, 116024 CHINA

PROFESSOR NAOTAKA IMADA (pp. 379-386)DEPARTMENT OF MATHEMATICSKANAZAWA MEDICAL UNIVERSITYUCHINADA, ISHIKAWA 920-02JAPAN

PROFESSOR ROGER V. JEANDEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCEUNIVERSITY OF QUEBEC AT RIMOUISKIRIMOUSKI, QUEBEC G5L 3AlCANADA

MR. DEREK JENNINGS4 BARFORD CLOSENORTH MILLERS DALECHANDLERS FORDHAMPSHIRE, S05 ITHENGLAND

PROFESSOR JAMES P. JONES (pp. 387-392)DEPARTMENT OF MATHEMATICS AND STATISTICSTHE UNIVERSITY OF CALGARY2500 UNIVERSITY DRIVE N.W.CALGARY, ALBERTACANADA T2N IN4

DR. A.W. AND PROFESSOR C.D. KEMPTHE MATHEMATICAL INSTITUTETHE UNIVERSITY OF ST ANDREWSST ANDREWSFIFE KY16 9SSSCOTLAND

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR ANN CHI KIM (pp. 393-398)DEPARTMENT OF MATHEMATICSTHE PUSAN NATIONAL UNIVERSITYDONGRAE, PUSAN 607, KOREAREPUBLIC OF KOREA

PROFESSOR WILLIAM A. KIMBALL (pp. 399-404)DEPARTMENT OF PURE AND APPLIED MATHEMATICSWASHINGTON STATE UNIVERSITYPULLMAN, WA 99164-2930

PROFESSOR CLARK KIMBERLING (pp. 405-416)DEPARTMENT OF MATHEMATICSUNIVERSITY OF EVANSVILLE1800 LINCOLN AVENUEEVANSVILLE, INDIANA 47722

DR. PETER KIRSCHENHOFER (pp. 263-272)INSTITUT FUR ALGEGRAUND DISKRETE MATHEMATIKTECHNISCHE UNIVERSITAT WIENWIEDNER HAUPTSTRASSE 8-10A-1040 WIEN, AUSTRIA

PROFESSOR DR. VIRGINIA S. KIRYAKOVA (pp. 159-168)BULGARIAN ACADEMY OF SCIENCESINSTITUTE OF MATHEMATICS1090 SOFIA, BULGARIA

DR. PETER KISS (pp. 417-420)ESZTERHAZY KAROLY TEACHERS'TRAINING COLLEGEDEPARTMENT OF MATHEMATICS3301 EGER PF.43, LEANYKA U. 4H-HUNGARY

DR. ARNOLD KNOPFMACHER (pp. 421-428)DEPARTMENT OF COMPUTER AND APPLIED MATHEMATICSUNIVERSITY OF THE WITWATERSRAND1 JAN SMUTS AVENUEJOHANNESBURG, WITS 2050SOUTH AFRICA

LT. COL. RICKEY A. KOLB (pp. 45-48)DEPARTMENT OF MATHEMATICSUNITED STATES MILITARY ACCADEMYWEST POINT, NY 10996-1786

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DR. JOSEPH LAHR56, RUE DE L'EGLISEL-7224 WALFERDANGEGRAND-DUCHY OF LUXEMBOURG

DR. WOLFDIETER LANG (pp. 429-440)INSTITUT FUR THEORETISCHE PHYSIKDER UNIVERSITAT KARLSRUHE(TH)KAISERSTRASSE 12PHYSIKHOCHHAUS 12, OGD-7500 KARLSRUHE 1, GERMANY

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR TIMOTHY LANGTRY (pp. 449-458)SCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF TECHNOLOGY, SYDNEYP.O. BOX 123BROADWAY N. S. W. 2007AUSTRALIA

DR. JACK Y. LEE (pp. 441-448)280 86ST., #D2BROOKLYN, NY 11209

PROFESSOR MORDECHAI LEWINDEPARTMENT OF MATHEMATICSTECHNION-ISRAEL INSTITUTE OF TECHNOLOGY32000 HAIFAISRAEL

PROFESSOR CALVIN T. LONG (pp. 449-458)DEPARTMENT OF PURE & APPLIED MATHEMATICSWASHINGTON STATE UNIVERSITYPULLMAN, WA 99164-3113

MR. FROSSO S. MAKRIDEPARTMENT OF MATHEMATICSUNIVERSITY OF PATRASPATRAS, GREECE

PROFESSOR C. D. MANNUNIVERSITY OF LONDONINTERNATIONAL HALLBRUNSWICK SQUARELONDON WC1N lASENGLAND

CONTRIBUTORS TO THIS PROCEEDINGS

DR. J. H. McCABETHE MATHEMATICS INSTITUTEUNIVERSITY OF ST. ANDREWSTHE NORTH HAUGHST. ANDREWS, FIFE KY16 9SSSCOTLAND

MR. RAY S. MELHAM3 DICKENSON STREETPANANIA N.S.W. 2213AUSTRALIA

MRS. JANE MILLER3401 LOWELL STREET N.W.WASHINGTON, DC 20016

PROFESSOR DR. WINFRIED B. MULLER (pp. 459-464)UNIVERSITY OF KLAGENFURTINSTITUTE OF MATHEMATICSUNIVERSITAETSSTRASSE 65-67A-9020 KLAGENFURT, AUSTRIA

DR. JOHN J. O'CONNORTHE MATHEMATICAL INSTITUTEUNIVERSITY OF ST. ANDREWSTHE NORTH HAUGHST. ANDREWS KY16 9SSFIFE, SCOTLAND

PROFESSOR R. L. OLLERTON (pp. 497-506)DEPARTMENT OF MATHEMATICSUNIVERSITY OF WESTERN SYDNEY, NEPEANNSW, 2747, AUSTRALIA

DR. ALAN OSWALD (pp. 459-464)SCHOOL OF COMPUTING & MATHEMATICSTEESSIDE POLYTECHNICMIDDLESBROUGHCLEVELAND TSI 3BAUNITED KINGDOM

PROFESSOR D. R. OWENS (pp. 497-506)UNIVERSITY OF WALESCOLLEGE OF MEDICINE, CARDIFFWALES, CF4 4XN, UNITED KINGDOM

PROFESSOR STAVROS G. PAPASTAVRIDIS (pp. 103-112)UNIVERSITY OF ATHENSMITSAKI35111 41 ATHENS, GREECE

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PROFESSOR RHODES PEELE (pp. 221-228; 229-244)DEPARTMENT OF MATHEMATICSAUBURN UNIVERSITY AT MONTGOMERYMONTGOMERY, AL 36193-0401

DR. S. P. PETHE (pp. 465-472)MATHEMATICS DEPARTMENTGOA UNIVERSITYGOA - 403 202. INDIA

DR. C. N. PHADTE (pp. 465-472)G.V.M.'S COLLEGE OFCOMMERCE AND ECONOMICSPONDA GOA 403 401INDIA

PROFESSOR ANDREAS N. PHILIPPOUGOVERNMENT HOUSE Z 50NICOSIA, CYPRUS

DR. GEORGE M. PHILLIPS (pp. 257-262)THE MATHEMATICAL INSTITUTEUNIVERSITY OF ST. ANDREWSTHE NORTH HAUGHST. ANDREWS KY16 9SSFIFE, SCOTLAND

MR. LOTHAR PIEPMEYER (pp. 279-288)FALLERSLEBER STRASSE 18D-W 3300 BRAUNSCHWEIGGERMANY

DR. JUKKA PIHKODEPARTMENT OF MATHEMATICSP.O. BOX 4 (HALLITUSKATU 15)SF-00014 UNIVERSITY OF HELSINKIH3LSINKI, FINLAND

PROFESSOR HELMUT PRODINGER (pp. 263-272; 473-480)INSTITUT FUR ALGEBRAUND DISKRETE MATHEMATIKTECHNISCHE UNIVERSITAT WIENWIEDNER HAUPTSTRASSE 8-10/118A-I040 WIEN, AUSTRIA

PROFESSOR NEVILLE ROBBINSDEPARTMENT OF MATHEMATICS1600 HOLLOWAY AVENUESAN FRANCISCO STATE UNIVERSITYSAN FRANCISCO, CA 94132

CONTRIBUTORS TO THIS PROCEEDINGS

MS. J. ADAIR ROBERTSON1204 WATAUGA STREETRALEIGH, NC 27604

DR. EDMUND F. ROBERTSON (pp. 77-84)UNIVERSITY OF ST ANDREWSDEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCESMATHEMATICS INSTITUTE, NORTH HAUGHST ANDREWS KY16 9SSFIFE, SCOTLAND

PROFESSOR ANDRZEJ ROTKIEWICZINSTYTUT MATEMATYCZNYPOLSKIEJ AKADEMII NAUK-PANSKR. SNIADECKICH 8P-00-950 WARSZAWA, POLAND

PROFESSOR SHUICHI SATO (pp. 481-486; 487-496)DEPARTMENT OF MATHEMATICSTSURUOKA TECHNICAL COLLEGEINO-OKA, TSURUOKAYAMAGATA 997, JAPAN

PROFESSOR DAIHACHIRO SATO (pp. 11-36; 37-44)DEPARTMENT OF MATHEMATICS & STATISTICSUNIVERSITY OF REGINAREGINA, SASKATCHEWANCANADA S4S OA2

DR. A. G. SCHAAKE (pp. 569-584; 585-600)DEPARTMENT OF MATHEMATICSWAIKATO POLYTECHNICHAMILTON, NEW ZEALAND

PROFESSOR A. G. SHANNON (pp. 331-344; 449-458; 497-506)SCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF TECHNOLOGY, SYDNEYP.O. BOX 123BROADWAY N. S. W. 2007AUSTRALIA

PROFESSOR DAVID B. SINGMASTERDEPARTMENT OF MATHEMATICSSOUTH BANK UNIVERSITYLONDON SEI OAA GREAT BRITAIN

DR. JOHN C. SJOBERG (pp. 507-514)105 CHARLES STREETCARLISLE, PA 17013

xxi

xxii

DR. S. A. SLEBARSKIDEPARTMENT OF PURE MATHEMATICSUNIVERSITY OF LEEDSLEEDSENGLAND LS2 9JT

CONTRIBUTORS TO THIS PROCEEDINGS

DR. GEOFF C. SMITH (pp. 49-60; 61-68; 245-256)SCHOOL OF MATHEMATICAL SCIENCESUNIVERSITY OF BATHCLAVERTON DOWN, BATH BA2 7AYENGLAND

DR. LAWRENCE SOMER (pp. 515-526; 527-546)DEPARTMENT OF MATHEMATICSTHE CATHOLIC UNIVERSITY OF AMERICAWASHINGTON, DC 20064

DR. LEA TERRACINI (pp. 547-560)UNIVERSITA DI TORINODEPARTIMENTO DI MATEMATICAVIA CARLO ALBERTO, 1010123 TORINO, ITALY

DR. RICHARD M. THOMAS (pp. 77-84)DEPARTMENT OF COMPUTING STUDIESUNIVERSITY OF LEICESTERUNIVERSITY ROADLEICESTER LEI 7RHENGLAND

PROFESSOR DR. ROBERT F. TICHY (pp. 263-272; 561-568)INSTITUT FUR MATH. 501 (A)TECHNISCHE UNIVERSITY, GRAZSTEYRERGASSE 30A-8010 GRAZ, AUSTRIA

MR. T. TSAPELAS (pp. 103-112)UNIVERSITY OF ATHENSDEPARTMENT OF MATHEMATICSSTATISTICAL UNITPANEPISTEMIOPOLIS, 15784ATHENS, GREECE

DR. JOHN C. TURNER (pp. 569-584; 585-600)DEPARTMENT OF MATHEMATICS AND STATISTICSUNIVERSITY OF WAIKATOPRIVATE BAG 3105HAMILTON, NEW ZEALAND

CONTRIBUTORS TO THIS PROCEEDINGS

PROFESSOR THERESA VAUGHANDEPARTMENT OF MATHEMATICSUNIVERSITY OF NORTH CAROLINAAT GREENSBOROGREENSBORO, NC 27412-5001

PROFESSOR MARCELLUS E. WADDILL (pp. 601-615)DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCEWAKE FOREST UNIVERSITYBOX 7388 REYNOLDA STATIONWINSTON-SALEM, NC 27109

PROFESSOR WILLIAM A. WEBB (pp. 399-404)DEPARTMENT OF PURE AND APPLIED MATHEMATICSWASHINGTON STATE UNIVERSITYPULLMAN, WA 99164-2930

xxiii

FOREWORD

This book contains 58 papers from among the 68 papers presented at the FifthInternational Conference on Fibonacci Numbers and Their Applications which was held at theUniversity of St. Andrews, St. Andrews, Fife, Scotland from July 20 to July 24, 1992. Thesepapers have been selected after a careful review by well known referees in the field, and theyrange from elementary number theory to probability and statistics. The Fibonacci numbers andrecurrence relations are their unifying bond.

It is anticipated that this book, like its four predecessors, will be useful to researchworkers and graduate students interested in the Fibonacci numbers and their applications.

June 5, 1993

xxv

The Editors

Gerald E. BergumSouth Dakota State UniversityBrookings, South Dakota, U.S.A.

Alwyn F. HoradamUniversity of New EnglandArmidale, N.S.W., Australia

Andreas N. PhilippouGovernment House Z50Nicosia, Cyprus

LOCAL COMMITTEE

THE ORGANIZING COMMITTEES

INTERNATIONAL COMMITTEE

Campbell, Colin M., Co-Chair

Phillips, George M., Co-Chair

Foster, Dorothy M.E.

McCabe, John H.

O'Connor, John J.

Robertson, Edmund F.

XXVll

Horadam, A.F. (Australia), Co-Chair

Philippou, A.N. (Cyprus), Co-Chair

Ando, S. (Japan)

Bergum, G.E. (U.S.A.)

Filipponi, P. (Italy)

Harborth, H. (Germany)

Johnson, M. (U.S.A.)

Kiss, P. (Hungary)

Long, C.T. (U.S.A.)

Popov, B.S. (Yugoslavia)

Turner, J. (New Zealand)

Waddill, M.E. (U.S.A.)

LIST OF CONTRIBUTORS TO THE CONFERENCE

*ALFORD, CECIL 0., (coauthor Daniel C. Fielder), "Contributions From CascadedCombinations to the Naming of Special Permutations."*ANDERSON, PETER G., "Multidimensional Golden Means."*ANDO, SHIRO, (coauthor Daihachiro Sato), "A Necessary And Sufficient Condition ThatRays Of A Star Configuration On Pascal's Triangle Cover Its Center With Respect To GCDAnd LCM."*ANDO, SHIRO, (coauthor Daihachiro Sato), "On The Minimal Center Covering Stars WithRespect To GCD In Pascal's Pyramid And Its Generalizations."*ARKIN, JOSEPH, (coauthors David C. Arney, Frank R. Giordano, Rickey A. Kolb and GeraldE. Bergum), "An Extension Of An Old Classical Diophantine Problem."ARKIN, JOSEPH, (coauthors David C. Arney, Frank R. Giordano and Rickey A. Kolb),"Stirling Relationships."*ARNEY, DAVID C., (coauthors Joseph Arkin, Frank R. Giordano, Rickey A. Kolb and GeraldE. Bergum), "An Extension Of An Old Classical Diophantine Problem."ARNEY, DAVID C., (coauthors Joseph Arkin, Frank R. Giordano and Rickey A. Kolb),"Stirling Relationships."*AYDIN, HUSEYIN, (coauthor Geoff C. Smith), "Fourier Analysis In Finite Nilpotent Groups."*AYDIN, HUSEYIN, (coauthors Ramszan Dikici, Geoff C. Smith) "Wall And VinsonRevisited."*BERGUM, GERALD E., (coauthors Joseph Arkin, David C. Arney, Frank R. Giordano andRickey A. Kolb), "An Extension Of An Old Classical Diophantine Problem."*BERGUM, GERALD E., (coauthor Charles K. Cook), "Integer Sided Triangles whose Ratio ofAltitude to Base is an Integer."*BICKNELL-JOHNSON, MARJORIE, "Designer Decimals: Fractions Which Contain SecondOrder Recursion Sequences In Their Decimal Expansions, Reading Left To Right Or Right ToLeft."BONG, NGUYEN HUU, "Fibonacci Numbers, largest Rock Polynomials and the MenageProblem."*CAMPBELL, COLIN M., (coauthors Edmund F. Robertson and Richard M. Thomas),"Semigroup Presentations And Number Sequences."*CERRUTI, UMBERTO, "Counting The Number Of Solutions Of Congruences."*CHRYSSAPHINOU, 0., (coauthors S. Papastavridis and T. Tsapelas), "On The NumberOverlapping Success Runs In A Sequence Of Independent Bernoulli Trials."*CHUAN, WAI-FONG, "Embedding Fibonacci Words into Fibonacci Word Patterns."*CLARY, STUART, (coauthor Paul D. Hemenway), "On Sums of Cubes of FibonacciNumbers."*COHEN, GRAEME L., (coauthors Calvin Long, Timothy Langtry and A. G. Shannon),"Arithmetic Sequences and Second Order Recurrences."*COOK, CHARLES K., (coauthor Gerald E. Bergum), "Integer Sided Triangles whose Ratio ofAltitude to Base is an Integer."*D'AMICO, ARNALDO, (coauthors Giuseppe Ferri and Marco Faccio), "The DFF and DFFzTriangles and Their Mathematical Properties."*DIKICI, RAMSZAN, (coauthors Huseyin Aydin, Geoff C. Smith) "Wall And VinsonRevisited."

*The asterisk indicates that the paper is included in this book.xxix

xxx CONTRIBUTORS TO THE CONFERENCE

*DILCHER, KARL, "On a Class of Iterative Recurrence Relations."*DIMOVSKI, IVAN H., (coauthor Virginia S. Kiryakova), "Discrete Operational Calculi forTwo-Sided Sequences."*DRESEL, L. A. G., "Transformations of Fibonacci-Lucas Identities."*DRMOTA, MICHAEL, "Fibonacci Numbers and Continued Fraction Expansions."*FACCIO, MARCO, (coauthors Giuseppe Ferri and Arnaldo D'Amico), "The DFF and DFFzTriangles and Their Mathematical Properties."*FERRI, GIUSEPPE, (coauthors Marco Faccio and Arnaldo D'Amico), "The DFF and DFFzTriangles and Their Mathematical properties."*FIELDER, DANIEL C., (coauthor Cecil O. Alford), "Contributions From CascadedCombinations to the Naming of Special Permutations."*FILIPPONI, PIERO, (coauthor Herta T. Freitag), "The Zeckendorf Representation of{Fkn/Fn}·"*FILIPPONI, PIERO, (coauthor Alwyn F. Horadam), "Integration Sequences of Fibonacci andLucas Polynomials." .*FLATH, DAN, (coauthor Rhodes Peele), "Fractal Patterns Derived From Rational BinomialCoefficients."*FLATH, DAN, (coauthor Rhodes'Peele), "Hausdorff Dimension in Pascal's Triangle."*FLETCHER, MATTHEW, (coauthor Geoff C. Smith), "Chaos, Elliptic Curves and All That."*FREITAG, H. T., (coauthor G. M. Phillips), "Co-Related Sequences Satisfying the GeneralSecond Order Recurrence Relation."*FREITAG, HERTA T., (coauthor Piero Filipponi), "The Zeckendorf Representation of{Fkn/Fn}·"*GARCIA, H., (coauthors J. C. Turner and A. G. Schaake), "Totient Functions on the EulerNumber Tree."*GIORDANO, FRANK R., (coauthors Joseph Arkin, David C. Arney, Rickey A. Kolb andGerald E. Bergum), "An Extension Of An Old Classical Diophantine Problem."GIORDANO, FRANK R., (coauthors David C. Arney, Joseph Arkin and Rickey A. Kolb),"Stirling Relationships."*GRABNER, P. J., (coauthors P. Kirschenhofer, H. Prodinger and R. F. Tichy), "On theMoments of the Sum-of-Digits Function."*HARBORTH, HEIKO, "Problems on Fibonacci and B-Adic Tree Representations by RegularK-Gons."*HARBORTH, HEIKO, (coauthor Lothar Piepmeyer), "Two-Distance Sets and the GoldenRatio."*HEMENWAY, PAUL D., (coauthor Stuart Clary), "On Sums of Cubes of FibonacciNumbers."*HENDEL, RUSSEL JAY, "A Fibonacci Problem Classification Scheme Useful toUndergraduate Pedagogy."*HORADAM, A. F., "Zeckendorf Representations of Positive and Negative Integers by PellNumbers."*HORADAM, Alwyn F., (coauthor Piero Filipponi), "Integration Sequences of Fibonacci andLucas Polynomials."*HORADAM, A. F., (coauthor A. G. Shannon), "Pell-Type Number Generators of PythagoreanTriples."*HORIBE, YASUICHI, "Balance Morphology of a Binary Tree."*HOWARD, F. T., "Norlund's Number B~nl."*HSU, LEETSCH C., "On Stirling-type Pairs and Extended Gegenbauer-Humbert-FibonacciPolynomials."*IMADA, NAOTAKA, "A Sequence Arising from Reflections in Multiple Glass Plates."JEAN, ROGER V., "Algorithms on Visible Pairs of Points in the Cylindrical Lattice."

CONTRIBUTORS TO THE CONFERENCE xxxi

*JONES, JAMES P., "Diophantine Representation of Non-Fibonacci Numbers."KEMP, A. W., "Second-Order Linear Recurrence Distributions and Their Properties."*KIM, ANN CHI, "On Some Problems Related to the Fibonacci Group."*KIMBALL, WILLIAM A., (coauthor William A. Webb) "Gongruence Properties of FibonacciNumbers and Fibonacci Coefficients Modulo p2."*KIMBERLING, CLARK, "Orderings of the Set of All Positive Fibonacci Sequences."*KIRSCHENHOFER, P., (coauthors P. J. Grabner, H. Prodinger and R. F. Tichy), "On theMoments of the Sum-of-Digits Function."*KIRYAKOVA, VIRGINIA S., (coauthor Ivan H. Dimovski), "Discrete Operational Calculi forTwo-Sided Sequences."*KISS, PETER, "Some Results Concerning the Reciprocal Sum of Prime Divisors of a LucasNumber."*KNOPFMACHER, ARNOLD, "Rational Numbers with Predictable Engel ProductExpansions."*KOLB, RICKEY A., (coauthors Joseph Arkin, David C. Arney, Frank R. Giordano and GeraldE. Bergum), "An Extension Of An Old Classical Diophantine Problem."KOLB, RICKEY A., (coauthors David C. Arney, Frank R. Giordano and Joseph Arkin),"Stirling Relationships."*LANG, WOLFDIETER, "Two Families of Orthogonal Polynomial Systems Related toFibonacci Chains."*LANGTRY, TIMOTHY, (coauthors Graeme L. Cohen, Calvin Long and A. G. Shannon),"Arithmetic Sequences and Second Order Recurrences."*LEE, JACK Y., "On the Formation of Higher Order Higher Dimensional Line-SequentialVector Spaces."LEWIN, MORDECHAI, "On Fibonacci and His Era"*LONG, CALVIN, (coauthors Graeme L. Cohen, Timothy Langtry and A. G. Shannon),"Arithmetic Sequences and Second Order Recurrences."MC BETH, MUHAMMAD ALI, (coauthor David Singmaster), "Generalized Diagonal Sums inPascal's Triangle."*MULLER, WINIFRIED B., (coauthor Alan Oswald), "Generalized Fibonacci Pseudoprimesand Probable Primes."*OLLERTON, R. 1., (coauthors A. G. Shannon and D. R. Owens), "A Cholesky Decompositionin Matching Insulin Profiles."*OSWALD, ALAN, (coauthor Winfried B. Muller), "Generalized Fibonacci Pseudoprimes andProbable Primes."*OWENS, D. R., (coauthors R. 1. Ollerton and A. G. Shannon), "A Cholesky Decomposition inMatching Insulin Profiles."*PAPASTAVRIDIS, S., (coauthors O. Chryssaphinou and T. Tsapelas), "On The NumberOverlapping Success Runs In A Sequence Of Independent Bernoulli Trials."*PEELE, RHODES, (coauthor Dan Flath), "Fractal Patterns Derived From Rational BinomialCoefficients."*PEELE, RHODES, (coauthor Dan Flath), "Hausdorff Dimension in Pascal's Triangle."*PETHE, S. P., (coauthor C. N. Phadte), "A Generalization of the Fibonacci Sequence."*PHADTE, C. N., (coauthor S. P. Pethe), "A Generalization of the Fibonacci Sequence."PHILIPPOU, ANDREAS, "Distributions of Order k and Generalized Fibonacci Polynomials"*PHILLIPS, G. M., (coauthor H. T. Freitag), "Co-Related Sequences Satisfying the GeneralSecond Order Recurrence Relation."*PIEPMEYER, LOTHAR, (coauthor Heiko Harborth), "Two-Distance Sets and the GoldenRatio."PIHKO, JUKKA, "An Elementarization of Bang's Proof of the Lucas-Lehmer Test."*PRODINGER, HELMUT, "How to Advance on a Stairway by Coin Flippings."

xxxii CONTRIBUTORS TO THE CONFERENCE

*PRODINGER, H., (coauthors P. Kirschenhofer, P. J. Grabner and R. F. Tichy), "On theMoments of the Sum-of-Digits Function."*ROBERTSON, EDMUND F., (coauthors Colin M. Campbell and Richard M. Thomas),"Semigroup Presentations And Number Sequences."ROTKIEWICZ, A., "Arithmetic Progressions formed by k Different Lehmer Pseudoprimes"*SATO, DAIHACHIRO, (coauthor Shiro Ando), "A Necessary And Sufficient Condition ThatRays Of A Star Configuration On Pascal's Triangle Cover Its Center With Respect To GCDAnd LCM."*SATO, DAIHACHIRO, (coauthor Shir~ Ando), "On The Minimal Center Covering Stars WithRespect To GCD In Pascal's Pyramid And Its Generalizations."*SATO, SHUICHI, "Fibonacci Sequence and Its Generalizations Hidden in Algorithms forGenerating Morse Codes."*SATO, SHUICHI, "On Matrix Representations of Generalized Fibonacci Numbers and TheirApplications."*SCHAAKE, A. G., (coauthor J. C. Turner), "The Elements of Enteger Geometry."*SCHAAKE, A. G., (coauthors H. Garcia and J. C. Turner), "Totient Functions on the EulerNumber-Tree."*SHANNON, A. G., (coauthors R. 1. Ollerton and D. R. Owens), "A Cholesky Decompositionin Matching Insulin Profiles."*SHANNON, A. G., (coauthor A. F. Horadam), "Pell-Type Number Generators of PythagoreanTriples."*SHANNON, A. G., (coauthors Graeme L. Cohen, Timothy Langtry and Calvin Long),"Arithmetic Sequences and Second Order Recurrences."SINGMASTER, DAVID, (coauthor Muhammad Ali McBeth), "Generalized Diagonal Sums inPascal's Triangle."*SJOBERG, JOHN C., "Generalized Exponential and Trigonometric Functions."*SMITH, GEOFF C., (coauthor Huseyin Aydin), "Fourier Analysis In Finite Nilpotent Groups."*SMITH, GEOFF C., (coauthors Ramszan Dikici, Huseyin Aydin) "Wall And VinsonRevisited."*SMITH, GEOFF C., (coauthor Matthew Fletcher), "Chaos, Elliptic Curves and All That."*SOMER, LAWRENCE, "Divisibility of Terms in Lucas Sequences by Their Subscripts."*SOMER, LAWRENCE, "Upper Bounds for Frequencies of Elements in Second-OrderRecurrences Over a Finite Field."*TERRACINI, LEA, "On the Convergence of Quotients of Some Recursive Sequences."*THOMAS, RICHARD M., (coauthors Edmund F. Robertson and Colin M. Campbell),"Semigroup Presentations And Number Sequences."*TICHY, ROBERT F., "Two Distribution Problems for Polynomials."*TICHY, R. F., (coauthors P. Kirschenhofer, H. Prodinger and P. J. Grabner), "On theMoments of the Sum-of-Digits Function."*TSAPELAS, T., (coauthors S. Papastavridis and O. Chryssaphinou), "On The Number OfOverlapping Success Runs In A Sequence Of Independent Bernoulli Trials."*TURNER, J. C., (coauthor A. G. Schaake), "The Elements of Enteger Geometry."*TURNER, J. C., (coauthors H. Garcia and A. G. Schaake), "Totient Functions on the EulerNumber Tree."*WADDILL, MARCELLUS E., "Using Matrix Techniques to Establish Properties of k-OrderLinear Recursive Sequences."*WEBB, WILLIAM A., (coauthor William A. Kimball) "Congruence Properties of FibonacciNumbers and Fibonacci Coefficients Modulo p2."

INTRODUCTION

The numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...,

known as the Fibonacci numbers, have been named by the nineteenth-century Frenchmathematician Edouard Lucas after Leonard Fibonacci of Pisa, one of the best .mathematiciansof the Middle Ages, who referred to them in his book Liber Abaci (1202) in connection with hisrabbit problem.

The astronomer Johann Kepler rediscovered the Fibonacci numbers, independently, andsince then several renowned mathematicians have dealt with them. We only mention a few: J.Binet, B. Lame, and E. Catalan. Edouard Lucas studied Fibonacci numbers extensively, andthe simple generalization

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...,

bears his name.

During the twentieth century, interest in Fibonacci numbers and their applications roserapidly. In 1961 the Soviet mathematician N. Vorobyov published Fibonacci Numbers, andVerner E. Hoggatt, Jr., followed in 1969 with his Fibonacci and Lucas Numbers. Meanwhile, in1963, Hoggatt and his associates founded The Fibonacci Association and began publishing TheFibonacci Quarterly. They also organized a Fibonacci Conference in California, U.S.A., eachyear for almost sixteen years until 1979. In 1984, the First International Conference onFibonacci Numbers and Their Applications was held in Patras, Greece, and the proceedings fromthis conference have been published. It was anticipated at that time that this conference wouldset the beginning of international conferences on the subject to be held every two or three yearsin different countries. With this intention as a motivating force, The Second InternationalConference on Fibonacci Numbers and Their Applications was held in San Jose, California,U.S.A., August 13-16, 1986. In order to carryon this new tradition, The Third InternationalConference on Fibonacci Numbers and Their Applications was held in Pisa, Italy, July 25-29,1988. The Fourth International Conference on Fibonacci Numbers and Their Applications washeld at Winston-Salem, North Carolina, July 3D-August 3, 1990. The proceedings from thesethree conferences have also been published. Because of the continuous success of the proceedingfour conferences, The Fifth International Conference on Fibonacci Numbers and TheirApplications was held at St. Andrews, Scotland July 20-24, and a Sixth Conference IS

scheduled for July 1994 in Pullman, Washington.

XXXlll

xxxiv INTRODUCTION

It is impossible to overemphasize the importance and relevance of the Fibonaccinumbers to the mathematical and physical sciences as well as other areas of study. TheFibonacci numbers appear in almost every branch of mathematics, obviously in number theory,but also in differential equations, probability, statistics, numerical analysis, and linear algebra.They also occur in physics, biology, chemistry, and electrical engineering.

It is believed that the contents of this book, like its predecessors, will prove useful toeveryone interested in this important branch of mathematics and that this material may lead toadditional results on Fibonacci numbers both in mathematics and in their applications to scienceand engineering.

The editors would like to acknowledge The Fibonacci Association, The EdinburghMathematical Society and The London Mathematical Society for their financial and otherassistance in making the Conference a success.

The Editors