In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.
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Polanco, Geremías, "Decomposition of Beatty and Complementary Sequences" (2023). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.