## Applications of Fibonacci Numbers, Volume 7G.E. Bergum, Andreas Philippou, Alwyn F. Horadam This book contains 50 papers from among the 95 papers presented at the Seventh International Conference on Fibonacci Numbers and Their Applications which was held at the Institut Fiir Mathematik, Technische Universitiit Graz, Steyrergasse 30, A-SOlO Graz, Austria, from July 15 to July 19, 1996. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers and recurrence relations are their unifying bond. It is anticipated that this book, like its six predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications. September 1, 1997 The Editors Gerald E. Bergum South Dakota State University Brookings, South Dakota, U. S. A. Alwyn F. Horadam University of New England Armidale, N. S. W. , Australia Andreas N. Philippou House of Representatives Nicosia, Cyprus xxvii THE ORGANIZING COMMITTEES LOCAL COMMITTEE INTERNATIONAL COMMITTEE Tichy, Robert, Chairman Horadam, A. F. (Australia), Co-Chair Prodinger, Helmut, Co-Chairman Philippou, A. N. (Cyprus), Co-Chair Grabner, Peter Bergurt:t, G. E. (U. S. A. ) Kirschenhofer, Peter Filipponi, P. (Italy) Harborth, H. (Germany) Horibe, Y. (Japan) Johnson, M. (U. S. A. ) Kiss, P. (Hungary) Phillips, G. M. (Scotland) Turner, J. (New Zealand) Waddill, M. E. (U. S. A. ) xxix LIST OF CONTRIBUTORS TO THE CONFERENCE *ADELBERG, ARNOLD, "Higher Order Bernoulli Polynomials and Newton Polygons. " AMMANN, ANDRE, "Associated Fibonacci Sequences. " *ANDERSON, PETER G. , "The Fibonacci Shuffle Tree. |

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### Contents

THE FIBONACCI SHUFFLE TREE | 9 |

GENERALIZATIONS TO LARGE HEXAGONS OF THE STAR OF DAVID THEOREM | 23 |

A NOTE ON A REPRESENTATION CONJECTURE BY HOGGATT | 39 |

Copyright | |

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1998 Kluwer Academic A.F. Horadam A.N. Philippou algebraic algorithm AMS Classification Numbers Applications of Fibonacci Bergum Bernoulli Polynomials binary tree binomial coefficients cells Cn(x coauthor column combinatorial Computer consider continued fraction Corollary defined denote derived sequence diophantine equation Diophantine quadruples Diophantus distribution of order divisor Edited by G.E. elements example F-code factors Fibonacci and Lucas Fibonacci Numbers Fibonacci Quarterly Fibonacci sequence Fibonacci tree Fibonacci vector Fibonacci-type polynomials finite follows formula function G. E. Bergum given Golden Section Hence hexagon Hoggatt identity infinite interval irreducible Kluwer Academic Publishers l(mod lattice Lemma linear recurring sequence Lucas numbers Lucas sequences Math Mathematics modulo multiplication Netherlands Newton polygon node notation number fields Number Theory obtain Pascal's triangle Pell numbers positive integer probable prime Proof Proposition proved pseudoprimes recurrence sequences respectively roots rule solution symmetric Theorem 3.2 values zero