论文标题
Zeckendorf游戏的界限
Bounds on Zeckendorf Games
论文作者
论文摘要
Zeckendorf证明，每个正整数$ n $都可以独特地写成非阳性斐波那契数的总和。我们使用这种分解来构建两人游戏。给定固定的整数$ n $和$ n = n f_1 $的初始分解，两个播放器通过使用与复发关系相关的动作$ f_ {n+1} = f_n+f_n+f_n+f_ {n-1} $交替进行交替。游戏总是在Zeckendorf分解中终止；根据运动长度的选择，并且获胜者可能会有所不同，尽管对于$ n \ ge 2 $，有一个非构造性证据，表明玩家2具有获胜策略。 最初，游戏长度的下限是订单$ n $（已知是尖锐的），而上限为$ n \ log n $。最近的工作将上限降低到尺寸$ n $，但其常数比猜想更大。我们改善上限并获得$ \ frac {\ sqrt {\ sqrt {5} +3} {2} {2} \ n -iz（n） - \ frac {1+ \ sqrt {5}} {2} {2} {2} z（n）$，是$ n $ z（n）$ n是$ n的$ n is $ n is of zeck n和z n $ n $ n $ n n n of zeck of zeck and z n n $ n n n n Nuger of the execk， $ iz（n）$是$ n $的zeckendorf分解中的索引总和（分别是$ \ log n $和$ \ log^2 n $）。我们还引入了一种贪婪的算法，该算法意识到上限，并表明，任何$ n $上最长的游戏都是通过尽可能应用分配动作来实现的。
Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer $n$ and an initial decomposition of $n=n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1}=F_n+F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition; depending on the choice of moves the length of the game and the winner can vary, though for $n\ge 2$ there is a non-constructive proof that Player 2 has a winning strategy. Initially the lower bound of the length of a game was order $n$ (and known to be sharp) while the upper bound was of size $n \log n$. Recent work decreased the upper bound to of size $n$, but with a larger constant than was conjectured. We improve the upper bound and obtain the sharp bound of $\frac{\sqrt{5}+3}{2}\ n - IZ(n) - \frac{1+\sqrt{5}}{2}Z(n)$, which is of order $n$ as $Z(n)$ is the number of terms in the Zeckendorf decomposition of $n$ and $IZ(n)$ is the sum of indices in the Zeckendorf decomposition of $n$ (which are at most of sizes $\log n$ and $\log^2 n$ respectively). We also introduce a greedy algorithm that realizes the upper bound, and show that the longest game on any $n$ is achieved by applying splitting moves whenever possible.
