Applications 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

N KLUWER ACADEMIC PUBLISHERS

DORDRECHT / BOSTON / LONDON

TABLE OF CONTENTS

A R E P O R T ON THE FIFTH INTERNATIONAL CONFERENCE ix LIST OF CONTRIBUTORS TO THIS PROCEEDINGS xi FOREWORD xxv THE ORGANIZING COMMITTEES xxvü LIST OF CONTRIBUTORS TO THE 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 CENTER WITH 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 CROUPS Huseyin Aydin and Geoff C. Smith 49

WALL AND VINSON REVISITED Huseyin Aydin, Ramazan Dikici and Geoff C. 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 NUMBER SEQUENCES

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

Umberto Cerruii 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 FOR TWO-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 0. Alford 207 THE ZECKENDORF REPRESENTATION OF {FkJFn}

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 Geoff C. 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 REGULÄR 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 NÖRLUND'S NUMBER B ^

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 P 2

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

Clark Kimberling 405

TABLE OF CONTENTS vn

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 HIGHER ORDER HIGHER DIMENSIONAL LINE-SEQUENTIAL VECTOR SPACES

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. Müller 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 BY THEIR SUBSCRIPTS

Lawrence Somer 515 UPPER BOUNDS FOR FREQUENCIES OF ELEMENTS IN SECOND-ORDER RECURRENCES OVER A 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 ENTEGER GEOMETRY

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

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