BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Faye Jackson (University of Michigan) DTSTART:20220527T200000Z DTEND:20220527T202500Z DTSTAMP:20260423T011437Z UID:CANT2022/56 DESCRIPTION:Title: The Generalized Bergman game\nby Faye Jackson (University of Michiga n) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAb stract\nP. Baird-Smith A. Epstein\, K. Flint\, and S. J. Miler\n(2018) cre ated the \\emph{Zeckendorf Game}\, a two-player game which takes\nas an in put a positive integer $n$ and\, using moves related to the\nFibonacci rec urrence relation\, outputs the unique decomposition of $n$\ninto a sum of non-consecutive Fibonacci numbers. Following this work and\nthat of G. Ber gman (1957)\, which proved the existence and uniqueness of\nsuch $\\varphi $-decompositions\, we formulate the \\emph{Bergman Game} which\noutputs th e unique decomposition of $n$ into a sum of non-consecutive\npowers of $\\ varphi$\, the golden mean.\n\nWe then formulate \\emph{Generalized Bergman Games}\, which use moves based\non an arbitrary non-increasing positive l inear recurrence relation and\noutput the unique decomposition of $n$ into a sum of non-adjacent powers\nof $\\beta$\, where $\\beta$ is the dominat ing root of the characteristic\npolynomial of the chosen recurrence relati on. We prove that the longest\npossible Generalized Bergman game on an ini tial state $S$ with $n$\nsummands terminates in $\\Theta(n^2)$ time\, and we also prove that the\nshortest possible Generalized Bergman game on an i nitial state terminates\nbetween $\\Omega(n)$ and $O(n^2)$ time. We also s how a linear bound on the\nmaximum length of the tuple used throughout the game.\n\nThis is joint work with Benjamin Baily\, Justine Dell\, Irfan Du rmic\, Henry\nFleischmann\, Isaac Mijares\, Steven J. Miller\, Ethan Pesik off\, Alicia Smith\nReina\, and Yingzi Yang.\n LOCATION:https://researchseminars.org/talk/CANT2022/56/ END:VEVENT END:VCALENDAR