Notes on Fibonacci Partitions
Notes on Fibonacci Partitions
复制标题
斐波那契分割的注意事项
DOI：
10.1080/10586458.2015.1118416
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发表时间：
2003
影响因子：
0.5
通讯作者：
F. Weinstein
中科院分区：
文献类型：
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作者：
F. Weinstein
摘要设f1 = 1, f2 = 2，且fi = fi−1 + fi−2，为Fibonacci数列。设Φh(n)为自然数n划分为h个不同斐波那契数的数量。根据n的Zeckendorf分割，我推导出了Φ（n; t）的一个公式，并且用它来分析F(n)的对象是Φ（n; 1）和χ(n)的对象是Φ（n;−1）。我得到了F(n)在fi−1≤n≤fi + 1−1时的最小上界。这意味着对于任意自然n。我也证明了|χ(n)|≥1，和。对于任何k小于2，我定义方程F(n) = k的特殊有限解集；所有的解都可以很容易地从。该构造使用有理数作为某些连分式的表示，并提供一个规范标识，其中Γ+是由< 1的正有理数自由生成的单似群。设Ψ(k)为的基数。我证明，对于I大于或等于2k和k大于或等于2，区间[fi−1，fi + 1−1]恰好包含方程F(n) = k的2Ψ(k)解，并提供序列Ψ(k)的Dirichlet生成函数的公式。我对方程F(n) = k随k变化的最小解集进行了推测，并提出了一些关于这些解的问题。
Abstract Let f1 = 1, f2 = 2, and fi = fi − 1 + fi − 2 for i > 2 be the sequence of Fibonacci numbers. Let Φh(n) be the quantity of partitions of natural number n into h different Fibonacci numbers. In terms of Zeckendorf partition of n, I deduce a formula for the function Φ(n; t) ≔ ∑h ⩾ 1Φh(n)th, and use it to analyze the functions F(n) ≔ Φ(n; 1) and χ(n) ≔ Φ(n; −1). I obtain the least upper bound for F(n) when fi − 1 ⩽ n ⩽ fi + 1 − 1. It implies that for any natural n. I prove also that |χ(n)| ⩽ 1, and . For any k ⩾ 2, I define a special finite set of solutions of the equation F(n) = k; all solutions can be easily obtained from . This construction uses a representation of rational numbers as certain continued fractions and provides with a canonical identification , where Γ+ is the monoid freely generated by the positive rational numbers < 1. Let Ψ(k) be the cardinality of . I prove that, for i ⩾ 2k and k ⩾ 2, the interval [fi − 1, fi + 1 − 1] contains exactly 2Ψ(k) solutions of the equation F(n) = k and offer a formula for the Dirichlet generating function of the sequence Ψ(k). I formulate conjectures on the set of minimal solutions of the equations F(n) = k as k varies and pose some questions concerning such solutions.