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Proceedings of ’The Fifth International Conference on Fibonacci Numbers and Their Applications’, The University of St. Andrews, Scotland, July 20July 24, 1992 PDF

625 Pages·1993·5.65 MB·English
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Preview Proceedings of ’The Fifth International Conference on Fibonacci Numbers and Their Applications’, The University of St. Andrews, Scotland, July 20July 24, 1992

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 1s t 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 ix LIST OF CONTRIBUTORS TO THIS PROCEEDINGS xi FOREWORD xxv THE ORGANIZING COMMITTEES xxvii LIST OF CONTRIBUTORS TOTHE CONFERENCE xxix INTRODUCTION xxxiii MULTIDIMENSIONAL GOLDEN MEANS Peter G. Anderson 1 A NECESSARY AND SUFFICIENT CONDITION THAT RAYS OF A STAR CONFIGURATION ON PASCAL'S TRIANGLE COVER ITS CENTERWITH RESPECT TO GCD AND LCM Shiro Ando and Daihachiro Sato 11 ON THE MINIMAL CENTER COVERING STARS WITH RESPECT TO GCD IN PASCAL'S PYRAMID AND ITS GENERALIZATIONS Shiro Ando and Daihachiro Sato 37 AN EXTENSION OF AN OLD CLASSICAL DIOPHANTINE PROBLEM Joseph Arkin, David C. Arney, Frank R. Giordano, Rickey A. Kolb and Gerald E. Bergum 45 FOURIER ANALYSIS IN FINITE NILPOTENT GROUPS Huseyin Aydin and GeoffC. Smith 49 WALL AND VINSON REVISITED Huseyin Aydin, Ramazan Dikici and GeoffC. Smith 61 DESIGNER DECIMALS: FRACTIONS WHICH CONTAIN SECOND ORDER RECURSION SEQUENCES IN THEIR DECIMAL EXPANSIONS, READING LEFT TO RIGHT OR RIGHT TO LEFT Marjorie Bicknell-Johnson 69 SEMIGROUP PRESENTATIONS AND NUMBERSEQUENCES Colin M. Campbell, Edmund F. Robertson and Richard M. Thomas 77 COUNTING THE NUMBEROF SOLUTIONS OF CONGRUENCES Umberto Cerruti 85 ON THE NUMBER OF OVERLAPPING SUCCESS RUNS IN A SEQUENCE OF INDEPENDENT BERNOULLI TRIALS O. Chryssaphinou, S. Papastavridis and T. Tsapelas 103 EMBEDDING FIBONACCI WORDS INTO FIBONACCI WORD PATTERNS Wai-fong Chuan 113 ON SUMS OF CUBES OF FIBONACCI NUMBERS Stuart Clary and Paul D. Hemenway 123 INTEGER SIDED TRIANGLES WHOSE RATIO OF ALTITUDE TO BASE IS AN INTEGER Charles K. Cook and Gerald E. Bergum 137 ON A CLASS OF ITERATIVE RECURRENCE RELATIONS Karl Dilcher 143 DISCRETE OPERATIONAL CALCULI FORTWO-SIDED SEQUENCES Ivan H. Dimovski and Virginia S. Kiryakova 159 TRANSFORMATIONS OF FIBONACCI-LUCAS IDENTITIES L. A. G. Dresel 169 FIBONACCI NUMBERS AND CONTINUED FRACTION EXPANSIONS Michael Drmota 185 v vi TABLE OF CONTENTS THE DFF AND DFFz TRIANGLES AND THEIR MATHEMATICAL PROPERTIES Marco Faccio, Giuseppe Ferri and Arnaldo D'Amico 199 CONTRIBUTIONS FROM CASCADED COMBINATIONS TO THE NAMING OF SPECIAL PERMUTATIONS Daniel C. Fielder and Cecil O. Alford 207 THE ZECKENDORF REPRESENTATION OF {Fkn/Fn} Piero Filipponi and Herta T. Freitag 217 FRACTAL PATTERNS DERIVED FROM RATIONAL BINOMIAL COEFFICIENTS Dan Flath and Rhodes Peele 221 HAUSDORFF DIMENSION IN PASCAL'S TRIANGLE Dan Flath and Rhodes Peele 229 CHAOS, ELLIPTIC CURVES AND ALL THAT Matthew Fletcher and GeoffC. Smith 245 CO-RELATED SEQUENCES SATISFYING THE GENERAL SECOND ORDER RECURRENCE RELATION H. T. Freitag and G. M. Phillips 257 ON THE MOMENTS OF THE SUM-OF-DIGITS FUNCTION P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy 263 PROBLEMS ON FIBONACCI AND B-ADIC TREE REPRESENTATIONS BY REGULAR K GONS Heiko Harborth 273 TWO-DISTANCE SETS AND THE GOLDEN RATIO Heiko Harborth and Lothar Piepmeyer 279 A FIBIONACCI PROBLEM CLASSIFICATION SCHEME USEFUL TO UNDERGRADUATE PEDAGOGY Russell Jay Hendel 289 ZECKENDORF REPRESENTATIONS OF POSITIVE AND NEGATIVE INTEGERS BY PELL NUMBERS A. F. Horadam 305 INTEGRATION SEQUENCES OF FIBONACCI AND LUCAS POLYNOMIALS Alwyn F. Horadam and Piero Filipponi 317 PELL-TYPE NUMBER GENERATORS OF PYTHAGOREAN TRIPLES A. F. Horadam and A. G. Shannon 331 BALANCE MORPHOLOGY OF A BINARY TREE Yasuichi Horibe 345 NORLUND'S NUMBER B(n) n F. T. Howard 355 ON STIRLING-TYPE PAIRS AND EXTENDED GEGENBAUER-HUMBERT-FIBONACCI POLYNOMIALS Leetsch C. Hsu 367 A SEQUENCE ARISING FROM REFLECTIONS IN MULTIPLE GLASS PLATES Naotaka Imada 379 DIOPHANTINE REPRESENTATION OF NON-FIBONACCI NUMBERS James P. Jones 387 ON SOME PROBLEMS RELATED TO THE FIBONACCI GROUP Ann Chi Kim 393 CONGRUENCE PROPERTIES OF FIBONACCI NUMBERS AND FIBONACCI COEFFICIENTS MODULO p2 William A. Kimball and William A. Webb 399 ORDERINGS OF THE SET OF ALL POSITIVE FIBONACCI SEQUENCES Clark Kimberling 405 TABLEOF CONTENTS VII SOME RESULTS CONCERNING THE RECIPROCAL SUM OF PRIME DIVISORS OF A LUCAS NUMBER Peter Kiss .417 RATIONAL NUMBERS WITH PREDICTABLE ENGEL PRODUCT EXPANSIONS Arnold Knopfmacher .421 TWO FAMILIES OF ORTHOGONAL POLYNOMIAL SYSTEMS RELATED TO FIBONACCI CHAINS Wolfdieter Lang .429 ON THE FORMATION OF HIGHERORDER HIGHER DIMENSIONAL LINE-SEQUENTIAL VECTORSPACES Jack Y. Lee 441 ARITHMETIC SEQUENCES AND SECOND ORDER RECURRENCES Calvin Long, Graeme L. Cohen, Timothy Langtry and A. G. Shannon .449 GENERALIZED FIBONACCI PSEUDOPRIMES AND PROBABLE PRIMES Winfried B. Muller and Alan Oswald .459 A GENERALIZATION OF THE FIBONACCI SEQUENCE S. P. Pethe and C. N. Phadte .465 HOW TO ADVANCE ON A STAIRWAY BY COIN FLIPPINGS Helmut Prodinger 473 FIBONACCI SEQUENCE AND ITS GENERALIZATIONS HIDDEN IN ALGORITHMS FOR GENERATING MORSE CODES Shuichi Sato 481 ON MATRIX REPRESENTATIONS OF GENERALIZED FIBONACCI NUMBERS AND THEIR APPLICATIONS Shuichi Sato 487 A CHOLESKY DECOMPOSITION IN MATCHING INSULIN PROFILES A. G. Shannon, R. L. Ollerton and D. R. Owens .497 GENERALIZED EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS John C. Sjoberg 507 DIVISIBILITY OF TERMS IN LUCAS SEQUENCES BYTHEIRSUBSCRIPTS Lawrence Somer 515 UPPER BOUNDS FOR FREQUENCIES OF ELEMENTS IN SECOND-ORDER RECURRENCES OVERA FINITE FIELD Lawrence Somer 527 ON THE CONVERGENCE OF QUOTIENTS OF SOME RECURSIVE SEQUENCES Lea Terracini 547 TWO DISTRIBUTION PROBLEMS FOR POLYNOMIALS Robert F. Tichy 561 THE ELEMENTS OF ENTEGERGEOMETRY J. C. Turner and A. G. Schaake 569 TOTIENT FUNCTIONS ON THE EULER NUMBERTREE J. C. Turner, H. Garcia and A. G. Schaake 585 USING MATRIX TECHNIQUES TO ESTABLISH PROPERTIES OF k-ORDER LINEAR RECURSIVE SEQUENCES Marcellus E. Waddill 601 SUBJECT INDEX 617 A REPORT ON THE FIFTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS Ever since our previous Meeting at Wake Forest University in North Carolina, the 1992 Conference had been awaited with keen anticipation. Finally, the announcement appeared: "sponsored jointly by The Fibonacci Association and The University of St. Andrews. THE FIFTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS will be held at The University of St. Andrews, Scotland, from July 20th to July 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, practically doubled previous attendances. All five continents were represented. From Europe there were 36; 29 came from America, 10 from Asia, 4 from Australia, and 1from Africa. Among the 24 countries represented by Conference participants, the United States provided the largest contingent 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 OUR INTERNATIONAL CONFERENCES," as he has initiated our FIRST meeting at Patras University in Greece in 1984. And in all our Conferences (and I do hope that in his proverbial modesty 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 doublesessions would become necessary. What caused the big increase in attendance? It may have been the fact that The University ofSt. Andrews is held in high esteem the world over. It may have been the magnetism, mathematical as well as personal, ofthe set ofco chairmen. Soul-searching choice decisions had to be madefor the overlapping sessions as there were 70 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" gave papers, 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 New Zealand). 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 the reputation 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 ofexistence, it has maintained vigorous scholarly impact across the wholeacademicspectrum. St. Andrews has been called "a gem of a University"-uniquely Scottish by history and beautiful location, yet unusually cosmopolitan. ix x A REPORT ON... The Conference's social events rounded off, and enhanced, our academic sessions. The traditional 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 quaintness of an historically rich palace and became enchanted by the charming multi-coloredness of the garden. To convey, the congenial and happy atmosphere at our Conference-dinner adequately would require a vocabulary far richer than mine. Interspersed with inspirational short talks and remarks, animated by delicious banquet fare and, most of all, by having our whole group gathered 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 facets of mathematics are duals of each other. And while we speak with many different accents, we understand each other on a much more significant level. Almost immediately, friendships blossomed or ripened, as the love of our discipline and the enthusiasm for it were written over all the faces of the "Fibonaccians" as some of us like to refer to ourselves. That one week in Scotland, kindled by the serenity of the Scottish landscape and enhanced by the spirit of our Scottish hosts and co-mathematicians, gave us experiences which were both mentally enriching and personally heartwarming. Finally, it was "farewell." But it is with much happiness that we can say: "Auf Wiedersehen in two yearsat Pullman, Washington." Herta T. Freitag LISTOF CONTRIBUTORS TO THIS PROCEEDINGS· PROFESSORCECIL O. ALFORD (pp. 207-216) SCHOOL OF ELECTRICAL ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY ATLANTA, GEORGIA 30332-0250 PROFESSOR PETERG. ANDERSON (pp. 1-10) SCHOOL OF COMPUTER SCIENCE AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY ONE LOMB MEMORIAL DRIVE POST OFFICE BOX 9887 ROCHESTER, NY 14623-0887 PROFESSORSHIRO ANDO (pp. 11-36; 37-44) COLLEGE OF ENGINEERING HOSEI UNIVERSITY 3-7-2, KAJINO-CHO KOGANEI-SHI, TOKYO 184 JAPAN MR. JOSEPH ARKIN (pp. 45-48) 197 OLD NYACK TURNPIKE SPRING VALLEY, NY 10977 COL. DAVID C. ARNEY (pp. 45-48) DEPARTMENT OF MATHEMATICS UNITED STATES MILITARY ACADEMY WEST POINT, NY 10996-1786 DR. HUSEYIN AYDIN (pp. 49-60; 61-68) ATATURK UNIVERSITY FEN-ED FACULTY DEPARTMENT OF MATHEMATICS ERZURUM,TURKEY DR. MICHEL BALLIEU 29 B5 PLACE MATTEOTTI B 1700 LA LOUVIERE BELGIUM .This list includes all authors and coauthors of papers presented at the conference even if their paper was rejected, published elsewhere or not submitted to the proceedings. Those who attended but did not present a paper are also in this list. XI

Description:
This book contains 58 papers from among the 68 papers presented at the Fifth International Conference on Fibonacci Numbers and Their Applications which was held at the University of St. Andrews, St. Andrews, Fife, Scotland from July 20 to July 24, 1992. These papers have been selected after a carefu
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