Proceedings of the Sixteenth International Conference on Fibonacci Numbers and Their Applications Rochester Institute of Technology, Rochester, New York, 2014 July 20{27 edited by Peter G. Anderson Rochester Institute of Technology Rochester, New York Christian Ballot Universit´eof Caen-Basse Normandie Caen, France William Webb Washington State University Pullman, Washington ii The Sixteenth Conference The 16th International Conference on Fibonacci Numbers and their Applica- tions was held on the large campus of the Rochester Institute of Technology, situated several miles off from downtown. It hosted about 65 participants from at least a dozen countries and all continents, northern Americans being most represented. Besides regular and occasional participants, there were a number of people who attended this conference for the first time. For instance, M´arton, 24, from Hungary, took three flights to reach Rochester; it was his first flying experiences, and we believe many appreciated his presence, and he himself en- joyed the whole package of the conference. These conferences are very congenial, being both scientific, social, and cultural events. This one had the peculiarity of having three exceptional presentations open to the public held on the Wednesday morning in a large auditorium filled with local young people and students, in addition to the conference participants. The Edouard´ Lucas invited lecturer, Jeffrey Lagarias, gave a broad well-applauded historic talk which ran from antiquity to present; Larry Ericksen, painter and mathematician, also had us travel through time and space, commenting on often famous artwork|okay, maybe the Golden ratio appeared a few too many times|and Arthur Benjamin, mathemagician (and mathematician) who, for some of his magics, managed the feat of both performing and explaining without loosing the audience a second. Peter Anderson was the grand host and organizer|at least two of the edi- tors wish to express their thanks to him|but many among the 65 participants will also remember fondly Ginny Gross-Abbey and Kim Shearer, for their un- compromising week-long help performed with competence and joy. We also offer our sincere thank you to the Rochester Institute of Technology, the B. Thomas Golisano College of Computing and Information Sciences, and its Dean Andrew Sears for generously hosting our meeting and providing us their superb facilities. A wine and cheese reception was held on the Sunday evening, which for many was their arrival date, an optional, memorable cruise and dinner on the Erie Canal was organized on the Tuesday; Wednesday afternoon was off with the possibility of visiting George Eastman's house, now an impressive museum. The conference banquet was held on the Thursday at a most original place, Artisan Works, surrounded by fine art. A day-trip to Niagara Falls took place on the last Saturday. iii iv Forword The sixteenth International Conference on Fibonacci Numbers and their Appli- cations was held in Rochester, New York, USA, July 20-26, on the campus of the Rochester Institute of Technology. This book contains 23 items which include a beautiful paper of Marjorie Bicknell-Johnson on the 50+ years of the existence of the Fibonacci Association, a compendium of problems posed, with occasional solutions, or partial solutions, found during the problem sessions and put together by Clark Kimberling, and 21 research articles from among the 49 papers and abstracts presented at the conference. These articles were selected and criticized by expert referees, whose time and care brought added value to this volume. Even though Fibonacci numbers and recurrences are the common bond to them all, the variety of topics and creativity of the papers compiled herein, is a testimony to the liveliness of this area of mathematics. It is our belief that the investigations reported in these proceedings, the 15th such proceedings emanating from this international conference, will stir up the curiosity of a number of researchers. Peter G. Anderson Christian Ballot William Webb v vi Contents Marjorie Bicknell-Johnson The Fibonacci Association: Historical Snapshots . 1 Clark Kimberling, Problem Editor Problem Proposals .......................... 5 Peter G. Anderson More Properties of the Zeckendorf Array . 15 Krassimir T. Atanassov, Daryl R. DeFord, and Anthony G. Shannon Pulsated Fibonacci Recurrences ................... 22 Arthur T. Benjamin and Elizabeth Reiland Combinatorial Proofs of Fibonomial Identities . 28 Andrew Best, Patrick Dynes, Xixi Edelsbrunner, Brian McDonald, Steven J. Miller, Kimsy Tor, Caroline Turnage-Butterbaugh, and Madeleine Weinstein Benford Behavior of Zeckendorf Decompositions . 35 Andrew Best, Patrick Dynes, Xixi Edelsbrunner, Brian McDonald, Steven J. Miller, Kimsy Tor, Caroline Turnage-Butterbaugh, and Madeleine Weinstein Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals . 47 Daniel Birmajer, Juan B. Gil, and Michael D. Weiner Convolutions of Tribonacci, Fuss-Catalan, and Motzkin Sequences 54 Jens-P. Bode and Heiko Harborth Steinhaus Triangles with Generalized Pascal Addition . 61 Minerva Catral, Pari Ford, Pamela Harris, Steven J. Miller, and Dawn Nelson Generalizing Zeckendorf's Theorem: The Kentucky Sequence . 68 Curtis Cooper Algebraic Statements Similar to Those in Ramanujan's \Lost Note- book" ................................. 91 Daryl DeFord Enumerating Distinct Chessboard Tilings . 102 M. Cetin Firengiz and Naim Tuglu On the q-Seidel Matrix . 117 Russell Jay Hendel A Cayley-Hamilton and Circulant Approach to Jump Sums . 124 vii Clark Kimberling and Peter J. C. Moses The Infinite Fibonacci Tree and Other Trees Generated by Rules 136 Takao Komatsu, Zuzana Mas´akov´a,and Edita Pelantov´a Higher-Order Identities for Fibonacci Numbers . 150 Florian Luca and Pantelimon St˘anic˘a On Fibonacci Numbers which are Elliptic Korselt Numbers . 164 Sam Northshield Three Analogues of Stern's Diatomic Sequence . 168 G. K. Panda and A. K. Panda Balancing-like Sequences Associated with Integral Standard Devi- ations of Consecutive Natural Numbers . 187 Pantelimon St˘anic˘a Normic Continued Fractions in Totally and Tamely Ramified Ex- tensions of Local Fields . 193 William Webb and Nathan Hamlin Compositions and Recurrences . 201 Paul Thomas Young Symmetries of Stirling Number Series . 205 Yifan Zhang and George Grossman Diophantine Triples and the Extendibility of 1; 2; 5 ; 1; 5; 10 . 212 Conference Participants f g f g . ....................................216 viii THE FIBONACCI ASSOCIATION: HISTORICAL SNAPSHOTS MARJORIE BICKNELL-JOHNSON Abstract. The Fibonacci Quarterly is now in its 52nd year of publication. V. E. Hoggatt, Jr., was editor for 18 years; G. E. Bergum, also editor for 18 years; and current editor, Curtis Cooper, 1998 to date. Marjorie Bicknell-Johnson was secretary of the Fibonacci Association from its beginning until 2010. This article gives a short history of the Fibonacci Association and some vignettes to bring that history to life. 1. Historical Snapshots and Rambles The first Fibonacci Quarterly, published in February 1963, had a subscription rate of $4.00 per year, and its Editor, Verner E. Hoggatt, Jr., held that position for L7 years. Vern’s friends told him the Quarterly wouldn’t last three years. Undaunted, he kept a mental list of “backsliders” who had not renewed their subscriptions and contacted each one personally. He persuaded, cajoled, and implored them so much that, in the end, it was impossible to say no. Vern corresponded and made friends with mathematicians from all over the world. He once hosted the world-famous mathematician Paul Erd¨osfor a month. Erd¨osarrived with one small suitcase, filled with silk underwear (because of allergies he was said to have). I met him, but he much preferred to talk to Vern. I found him strange; he had no home, no family, and all of his possessions were in his suitcase. I think he found me strange as well, visiting a college professor with my two little “epsilons,” the Erd¨osword for children. Erd¨osand Vern worked on dozens of problems during that time and wrote one paper [6] together, which I typed. As many of you may know, it was considered a great honor to have published a paper with Erd¨os, so much so that on Google, one can find the Erd¨osnumber of those who wrote with Erd¨os: 1; those who wrote with someone who wrote with Erd¨os:2; and so on. So, Vern has Erd¨os number 1, while Jerry Bergum and I each have Erd¨osnumber 2. The Managing Editor, Brother Alfred Brousseau, who was equally positive and enthusiastic about anything dealing with Fibonacci numbers, typed the first issue of the Quarterly and kept track of subscriptions and the bank account. He played the accordion and loved to lead group singing. In tune with his personality, he wrote the ballad, “Do What Comes Fibonaturally,” to the melody of “The Blue-Tail Fly.” Additionally, he compiled a bibliography of 700 Fibonacci references ranging from recreational to serious research, quite a feat in pre-computer times, and collected cones from every species of pine found in California to illustrate the Fibonacci patterns found in spirals of their scales. He inspired Vern to grow a large sunflower, his so-called Lucas sunflower, which had 76 clockwise spirals and 47 counterclockwise ones. As one of Professor Hoggatt’s students, I came on board in 1962. He was well liked by students at San Jose State College; they nicknamed him “Professor Fibonacci,” and I soon found out why: he took any and every opportunity to lecture on the Fibonacci sequence. We all loved it. 1 THE FIBONACCI QUARTERLY Dr. Hoggatt—I called him that as his student—brought unusual problems to us for home- work. A curiosa in Scripta Mathematica [7] claimed that, if the nine positive digits are ar- ranged in a square array, the non-negative determinant values range from 0 through 512 but with a couple dozen missing values. So we were each assigned to write and evaluate 20 such determinants—find those missing values, or show that they were indeed impossible.