The Generalized Bergman Game
Abstract
Every positive integer may be written uniquely as a base$\beta$ decompositionthat is a legal sum of powers of $\beta$where $\beta$ is the dominating root of a nonincreasing positive linear recurrence sequence. Guided by earlier work on a twoplayer game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of twoplayer games played on an infinite tuple of nonnegative integers which decompose a positive integer into its base$\beta$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $\Theta(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $\Omega(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2109.00117
 Bibcode:
 2021arXiv210900117B
 Keywords:

 Mathematics  Number Theory;
 11P99 (Primary);
 11K99 (Secondary)
 EPrint:
 34 pages, 6 figures, to be submitted in Fibonacci Quartlerly