Applications of Fibonacci Numbers Volume 5

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 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 four predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications. June 5, 1993 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 Government House Z50 Nicosia, Cyprus xxv THE ORGANIZING COMMITTEES LOCAL COMMITTEE INTERNATIONAL COMMITTEE Campbell, Colin M., Co-Chair Horadam, A.F. (Australia), Co-Chair Phillips, George M., Co-Chair Philippou, A.N. (Cyprus), Co-Chair Foster, Dorothy M.E. Ando, S. (Japan) McCabe, John H. Bergum, G.E. (U.S.A.) Filipponi, P. (Italy) O'Connor, John J.

Multidimensional Golden Means.- A Necessary and Sufficient Condition that Rays of A Star Configuration on Pascal’s Triangle Cover Its Center With Respect to GCD and LCM.- On the Minimal Center Covering Stars With Respect to GCD In Pascal’s Pyramid and Its Generalizations.- An Extension of an Old Classical Diophantine Problem.- Fourier Analysis in Finite Nilpotent Groups.- Wall and Vinson Revisited.- Designer Decimals: Fractions Which Contain Second Order Recursion Sequences in Their Decimal Expansions, Reading Left to Right or Right to Left.- Semigroup Presentations and Number Sequences.- Counting the Number of Solutions of Congruences.- On the Number of Overlapping Success Runs in A Sequence of Independent Bernoulli Trials.- Embedding Fibonacci Words Into Fibonacci Word Patterns.- On Sums of Cubes of Fibonacci Numbers.- Integer Sided Triangles Whose Ratio of Altitude to Base Is an Integer.- On a class of Iterative Recurrence Relations.- Discrete Operational Calculi for Two-Sided Sequences.- Transformations of Fibonacci-Lucas Identities.- Fibonacci Numbers and Continued Fraction Expansions.- The DFF and DFFZ Triangles and Their Mathematical Properties.- Contributions from Cascaded Combinations to the Naming of Special Permutations.- The Zeckendorf Representation of {Fkn/Fn}.- Fractal Patterns Derived from Rational Binomial Coefficients.- Hausdorff Dimension in Pascal’s Triangle.- Chaos, Elliptic Curves and All that.- Co-Related Sequences Satisfying the General Second Order Recurrence Relation.- On the Moments of The Sum-of-Digits Function.- Problems on Fibonacci and B-Adic Tree Representations by Regular K-Gons.- Two-Distance Sets and the Golden Ratio.- A Fibionacci Problem Classification Scheme Useful to Undergraduate Pedagogy.- Zeckendorf Representations of Positive and Negative Integers by Pell Numbers.- Integration Sequences of Fibonacci and Lucas Polynomials.- Pell-Type Number Generators of Pythagorean Triples.- Balance Morphology of a Binary Tree.- Nörlund’s Number B(n)n.- On Stirling-Type Pairs and Extended Gegenbauer-Humbert-Fibonacci Polynomials.- A Sequence Arising from Reflections in Multiple Glass Plates.- Diophantine Representation of Non-Fibonacci Numbers.- On Some Problems Related to the Fibonacci Group.- Congruence Properties of Fibonacci Numbers and Fibonacci Coefficients Modulo P2.- Orderings of the Set of All Positive Fibonacci Sequences.- Some Results Concerning the Reciprocal Sum of Prime Divisors of a Lucas Number.- Rational Numbers with Predictable Engel Product Expansions.- Two Families of Orthogonal Polynomial Systems Related to Fibonacci Chains.- On the Formation of Higher Order Higher Dimensional Line-Sequential Vector Spaces.- Arithmetic Sequences and Second Order Recurrences.- Generalized Fibonacci Pseudoprimes and Probable Primes.- A Generalization of the Fibonacci Sequence.- How to Advance on a Stairway by Coin Flippings.- Fibonacci Sequence and Its Generalizations Hidden in Algorithms for Generating Morse Codes.- On Matrix Representations of Generalized Fibonacci Numbers and Their Applications.- A Cholesky Decomposition in Matching Insulin Profiles.- Generalized Exponential and Trigonometric Functions.- Divisibility of Terms in Lucas Sequences by Their Subscripts.- Upper Bounds for Frequencies of Elements in Second-Order Recurrences Over a Finite Field.- On the Convergence of Quotients of Some Recursive Sequences.- Two Distribution Problems for Polynomials.- The Elements of Enteger Geometry.- Totient Functions on the Euler Number Tree.- Using Matrix Techniques to Establish Properties of k-Order Linear Recursive Sequences.