Article

The Bergman Game

Authors:
Benjamin Baily
Benjamin Baily
Benjamin Baily
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Justine Dell at Haverford College
Justine Dell
Irfan Durmić at University of Jyväskylä
Irfan Durmić
Henry Fleischmann at University of Michigan
Henry Fleischmann
Show all 11 authorsHide
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

View
The Accelerated Zeckendorf Game
Article
  • Sep 2024
  • Fibonacci Q
The Zeckendorf decomposition of a positive integer n is the unique set of non-consecutive Fibonacci numbers that sum to n. Baird-Smith et al. defined a game on Fibonacci decompositions of n, called the Zeckendorf Game. This paper introduces a variant of the Zeckendorf Game, called the Accelerated Zeckendorf Game, where a player may play as many moves of the same type as possible on their turn. We prove that a sharp lower bound on the game length of the Accelerated Zeckendorf Game is k−1k-1, where k is the index of the largest term in the Zeckendorf decomposition of n. We conjecture that Player 1 has a winning strategy if n>9n>9. We conjecture that the distribution of game lengths tends to a Gaussian as n goes to infinity, and that the average game length grows sub-linearly in n.
View
The Generalized Zeckendorf Game
Preprint
Full-text available
  • Sep 2018
Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use these decompositions to construct a two-player game, which can be completely analyzed for linear recurrence relations of the form Gn=∑i=1kcGn−iG_n = \sum_{i=1}^{k} c G_{n-i} for a fixed positive integer c (c=k−1=1c=k-1=1 gives the Fibonaccis). Given a fixed integer n and an initial decomposition of n=nG1n = n G_1, the two players alternate by using moves related to the recurrence relation, and whomever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible; for the Fibonacci game the upper bound is on the order of nlog⁡nn\log n, and for other games we obtain a bound growing linearly with n. For the Fibonacci game, Player 2 has the winning strategy for all n>2n > 2. If Player 2 makes a mistake on his first move, however, Player 1 has the winning strategy instead. Interestingly, the proof of both of these claims is non-constructive.
View
Finite beta-expansions
Article
Full-text available
  • Dec 1992
  • ERGOD THEOR DYN SYST
We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.
View
View
View
View
The Zeckendorf Game
Chapter
  • Jan 2020
Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this to create a two-player game. Given a fixed integer n and an initial decomposition of n=nF1n = n F_1, the two players alternate by using moves related to the recurrence relation Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1}, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible. The upper bound is on the order of nlog⁡nn\log n, and the lower bound is sharp at n−Z(n)n-Z(n) moves, where Z(n) is the number of terms in the Zeckendorf decomposition of n. Notably, Player 2 has the winning strategy for all n>2n > 2; interestingly, however, the proof is non-constructive.
View
Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions
Article
  • Jun 2016
Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers {1,2,3,5,… }\{1,2,3,5,\dots\}. This has been extended to many other recurrence relations {Gn}\{G_n\} (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an m∈[Gn,Gn+1)m \in [G_n, G_{n+1}) converges to a Gaussian as n→∞n\to\infty. We prove that for any non-negative integer g the average number of gaps of size g in many generalized Zeckendorf decompositions is Cμn+dμ+o(1)C_\mu n+d_\mu+o(1) for constants Cμ>0C_\mu > 0 and dμd_\mu depending on g and the recurrence, the variance of the number of gaps of size g is similarly Cσn+dσ+o(1)C_\sigma n + d_\sigma + o(1) with Cσ>0C_\sigma > 0, and the number of gaps of size g of an m∈[Gn,Gn+1)m\in[G_n,G_{n+1}) converges to a Gaussian as n→∞n\to\infty. The proof is by analysis of an associated two-dimensional recurrence; we prove a general result on when such behavior converges to a Gaussian, and additionally re-derive other results in the literature.
View
Symbolic dynamics of (-β)-expansions
Article
  • Jan 2012
We study shift spaces associated with a family of transformations generating ex-pansions in base -β, with β >1. We give a complete characterization when these shift spaces are sofic or even shifts of finite type.
View
View
View
View
The Distribution of Gaps between Summands in Generalized Zeckendorf Decompositions
Article
  • Feb 2014
Zeckendorf proved that any integer can be decomposed uniquely as a sum of non-adjacent Fibonacci numbers, FnF_n. Using continued fractions, Lekkerkerker proved the average number of summands of an m∈[Fn,Fn+1)m \in [F_n, F_{n+1}) is essentially n/(φ2+1)n/(\varphi^2 +1), with φ\varphi the golden ratio. Miller-Wang generalized this by adopting a combinatorial perspective, proving that for any positive linear recurrence the number of summands in decompositions for integers in [Gn,Gn+1)[G_n, G_{n+1}) converges to a Gaussian distribution. We prove the probability of a gap larger than the recurrence length converges to decaying geometrically, and that the distribution of the smaller gaps depends in a computable way on the coefficients of the recurrence. These results hold both for the average over all m∈[Gn,Gn+1)m \in [G_n, G_{n+1}), as well as holding almost surely for the gap measure associated to individual m. The techniques can also be used to determine the distribution of the longest gap between summands, which we prove is similar to the distribution of the longest gap between heads in tosses of a biased coin. It is a double exponential strongly concentrated about the mean, and is on the order of log⁡n\log n with computable constants depending on the recurrence.
View
Generalized Zeckendorf expansions
Article
  • Mar 1994
  • APPL MATH LETT
Digital expansions with respect to linear recurring sequences are considered. Using a general result due to Frougny and Solomyak, finite representations are investigated and a quantitative refinement is established.
View
β-Expansions and symbolic dynamics
Article
  • Jun 1989
  • THEOR COMPUT SCI
Given a real number β > 1, it is easy to expand x ϵ [0, 1] in a “canonical” way into a series of negative powers of β with integer coefficients less than or equal to [β]. The form of the set of such “β-expansions”—which is a matter of symbolic dynamics or language theory—varies according to arithmetical properties of β, but there are many common features. Known facts (which are few) and open questions (much more numerous) are reviewed in this paper.
View
From Fibonacci Numbers to Central Limit Type Theorems
Article
  • Aug 2010
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1∞\{F_n\}_{n=1}^{\infty}. Lekkerkerker proved that the average number of summands for integers in [Fn,Fn+1)[F_n, F_{n+1}) is n/(ϕ2+1)n/(\phi^2 + 1), with ϕ\phi the golden mean. This has been generalized to the following: given nonnegative integers c1,c2,...,cLc_1,c_2,...,c_L with c1,cL>0c_1,c_L>0 and recursive sequence {Hn}n=1∞\{H_n\}_{n=1}^{\infty} with H1=1H_1=1, Hn+1=c1Hn+c2Hn−1+...+cnH1+1H_{n+1} =c_1H_n+c_2H_{n-1}+...+c_nH_1+1 (1≤n<L)(1\le n< L) and Hn+1=c1Hn+c2Hn−1+...+cLHn+1−LH_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L} (n≥L)(n\geq L), every positive integer can be written uniquely as ∑aiHi\sum a_iH_i under natural constraints on the aia_i's, the mean and the variance of the numbers of summands for integers in [Hn,Hn+1)[H_{n}, H_{n+1}) are of size n, and the distribution of the numbers of summands converges to a Gaussian as n goes to the infinity. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to a multitude of other problems (in the sequel paper \cite{BM} we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the ±Fn\pm F_n's, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely −(21−2ϕ)/(29+2ϕ)≈−0.551058-(21-2\phi)/(29+2\phi) \approx -0.551058.
View