Table Of ContentApplications 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
xii CONTRIBUTORS TO THIS PROCEEDINGS
PROFESSORGERALD E. BERGUM (pp. 45-48; 137-142)
COMPUTERSCIENCE DEPARTMENT
SOUTH DAKOTA STATE UNIVERSITY
BOX 2201
BROOKINGS, SD 57007-0194
DR. MARJORIE BICKNELL-JOHNSON (pp. 69-76)
665 FAIRLANE AVENUE
SANTA CLARA, CA 95051
PROFESSOR NGUYEN-HUU BONG
DEPARTMENT OF MATHEMATICS
UNIVERSITI BRUNEI DARUSSALAM
GADONG 3186 BRUNEI
DR. RUTH BOWER
525-55TH STREET
WEST PALM BEACH, FL 33407-2607
PROFESSOR A. V. BOYD
DEPARTMENT OF STATISTICS & ACTUARIAL SCIENCE
UNIVERSITY OF THE WITWATERSRAND
1JAN SMUTS AVENUE
JOHANNESBURG 2050, SOUTH AFRICA
DR. COLIN M. CAMPBELL (pp. 77-84)
UNIVERSITY OF ST ANDREWS
DEPARTMENT OF MATHEMATICS AND COMPUTERSCIENCE
MATHEMATICS INSTITUTE, NORTH HAUGH
ST ANDREWS KY16 9SS
FIFE, SCOTLAND
PROFESSOR UMBERTO CERRUTI (pp. 85-102)
UNIVERSITA DI TORINO
DIPARTIMENTO DI MATEMATICA
VIA PRINCIPE AMEDEO 8
10123 TORINO, ITALIA
DR. OURANIA CHRYSSAPHINOU (pp. 103-112)
UNIVERSITY OF ATHENS
DEPARTMENT OF MATHEMATICS
STATISTICAL UNIT
PANEPISTEMIOPOLIS, 15784
ATHENS, GREECE
PROFESSORWAI-FONG CHUAN (pp. 113-122)
DEPARTMENT OF MATHEMATICS
CHUNG-YUAN CHRISTIAN UNIVERSITY
CHUNG LI, TAIWAN 32023
REPUBLIC OF CHINA
