Generalizing Zeckendorf's Theorem to f-decompositions
Generalizing Zeckendorf's Theorem to f-decompositions
复制标题
将 Zekendorf 定理推广到 f 分解
DOI：
10.1016/j.jnt.2014.01.028
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发表时间：
2013
影响因子：
0.7
通讯作者：
Umang Varma
中科院分区：
文献类型：
--
作者：
Philippe Demontigny;Thao T. Do;Archit Kulkarni;Steven J. Miller;David Moon;Umang Varma
Zeckendorf的一个美丽的定理指出，每个正整数都可以唯一地分解为非连续Fibonacci数{F n}的和，其中F 1= 1，F 2= 2，F n+ 1= F n+ F n− 1。对于一般递归{G n}与非负系数，有一个概念的法律的分解，再次导致一个独特的表示。我们考虑匡威的问题：给定一个法律的分解的概念，构造一个序列{a n}，使得每个正整数都可以唯一地分解为a n的和。我们证明这是可能的一个概念的法律的分解称为f-分解。这个概念推广了现有的概念，如base-b表示，Zeckendorf分解和阶乘数系统。使用这个新的视角，我们扩大了Zeckendorf型结果的范围，概括了以前的研究范围。最后，对于特定类别的概念的分解，我们证明了高斯结果关于分布的被加数在分解的随机选择的整数。视频有关本文的视频摘要，请点击此处或访问http：//youtu。是/hnYJwvOfzLo。
Text A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers {F n}, where F 1= 1, F 2= 2 and F n+ 1= F n+ F n− 1. For general recurrences {G n} with nonnegative coefficients, there is a notion of a legal decomposition which again leads to a unique representation. We consider the converse question: given a notion of legal decomposition, construct a sequence {a n} such that every positive integer can be uniquely decomposed as a sum of a n's. We prove this is possible for a notion of legal decomposition called f-decompositions. This notion generalizes existing notions such as base-b representations, Zeckendorf decompositions, and the factorial number system. Using this new perspective, we expand the range of Zeckendorf-type results, generalizing the scope of previous research. Finally, for specific classes of notions of decomposition we prove a Gaussianity result concerning the distribution of the number of summands in the decomposition of a randomly chosen integer. Video For a video summary of this paper, please click here or visit http://youtu. be/hnYJwvOfzLo.