论文标题
正线性复发序列的完整性
Completeness of Positive Linear Recurrence Sequences
论文作者
论文摘要
如果每个正整数都是不同术语的总和，则一系列正整数是完整的。正线性复发序列（PLR）是一个序列，该序列是由形式$ h_ {n + 1} = c_1 H_n + \ cdots + c_l H_l H_ {N-l + 1} $ h_1 h_n + 1} = c_1 H_n + 1}的均质线性复发关系定义的。 我们寻求通过完整性对各种PLR进行分类。通过修改PLR的复发系数如何影响完整性的结果，我们完全表征了PLR的几个家族的完整性以及对更多一般家庭的猜测标准。我们的主要方法是应用Brown的标准，该标准说，增加的序列$ \ {h_n \} _ {n = 1}^{\ infty} $是完整的，并且仅当$ h_1 = 1 $ and $ h_1 = 1 $和$ h_ {n + 1} \ leq 1 + sum_ = 1 + \ sum_ = 1} i = 1} n h__i $。 ％对这些结果的调查可以在\ cite {bhllmt}中找到。 最后，我们采用了PLRS的先前分析工作，以找到一种更有效的检查完整性的方法。具体而言，任何PLR的特征多项式都具有一个正词。通过界定该根的大小，大多数序列可以分类为完整或不完整。此外，我们表明存在一个不确定的区域，其中主要根没有揭示有关完整性的任何信息。我们已经猜想了该区域的精确界限。
A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n-L+1}$ and a particular set of initial conditions. We seek to classify various PLRS's by completeness. With results on how completeness is affected by modifying the recurrence coefficients of a PLRS, we completely characterize completeness of several families of PLRS's as well as conjecturing criteria for more general families. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. %A survey of these results can be found in \cite{BHLLMT}. Finally, we adopt previous analytic work on PLRS's to find a more efficient way to check completeness. Specifically, the characteristic polynomial of any PLRS has exactly one positive root; by bounding the size of this root, the majority of sequences may be classified as complete or incomplete. Additionally, we show there exists an indeterminate region where the principal root does not reveal any information on completeness. We have conjectured precise bounds for this region.
