Average number of Zeckendorf integers

Text: By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence, and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences {G_n} that have the same linear recurrence relations as the Nth order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G_n. We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video: For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns.