论文标题
正线性复发序列的完整性简介
An Introduction to Completeness of Positive Linear Recurrence Sequences
论文作者
论文摘要
正线性复发序列（PLR）是由带正系数和特定初始条件集的均质线性复发关系定义的序列。如果每个正整数都是序列的不同术语的总和，则一系列正整数是\ emph {complete}。 Zeckendorf定理的结果之一是斐波那契数的序列已完成。以前的工作已经为所有PLRS建立了广义的Zeckendorf定理。我们考虑PLRS的内容，并希望将它们分类为完整。我们研究如何通过修改PLR的复发系数来影响完整性。然后，我们在许多情况下确定哪些序列是由表格$ [1，\ ldots，1，0，\ ldots，0，n] $完成的序列。此外，我们猜想了其他PLRS家族中其他序列的其他最大最后系数的界限。我们的主要方法是应用Brown的标准，该标准说，增加的序列$ \ {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 $。本文是对该主题的介绍，在正线性复发序列ARXIV的完整性中进一步探讨了ARXIV：2010.01655。
A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if every positive integer is a sum of distinct terms of the sequence. One consequence of Zeckendorf's theorem is that the sequence of Fibonacci numbers is complete. Previous work has established a generalized Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify them as complete or not. We study how completeness is affected by modifying the recurrence coefficients of a PLRS. Then, we determine in many cases which sequences generated by coefficients of the forms $[1, \ldots, 1, 0, \ldots, 0, N]$ are complete. Further, we conjecture bounds for other maximal last coefficients in complete sequences in other families of PLRS's. 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 $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. This paper is an introduction to the topic that is explored further in Completeness of Positive Linear Recurrence Sequences arXiv:2010.01655.
