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A positive linear recurrence sequence (PLRS) is a sequence defined by ahomogeneous linear recurrence relation with positive coefficients and aparticular set of initial conditions. A sequence of positive integers is\emph{complete} if every positive integer is a sum of distinct terms of thesequence. One consequence of Zeckendorf's theorem is that the sequence ofFibonacci numbers is complete. Previous work has established a generalizedZeckendorf's theorem for all PLRS's. We consider PLRS's and want to classifythem as complete or not. We study how completeness is affected by modifying therecurrence coefficients of a PLRS. Then, we determine in many cases whichsequences generated by coefficients of the forms $[1, \ldots, 1, 0, \ldots, 0,N]$ are complete. Further, we conjecture bounds for other maximal lastcoefficients in complete sequences in other families of PLRS's. Our primarymethod 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 thatis explored further in Completeness of Positive Linear Recurrence SequencesarXiv:2010.01655.

An Introduction to Completeness of Positive Linear Recurrence Sequences