OEIS A000062: BD sequences, complementary partitions, and Sturmian connections

Join us as we unpack A000062, the Beatty-type BD sequence given by floor(n/(e−2)). We’ll explore its complementary partner A194807, Rayleigh’s BD theorem, and how the pair partitions the positive integers. We’ll also glimpse the deep links to Sturmian words (the first differences form a characteristic Sturmian word) and even a surprising musical connection: the black/white-key pattern of a piano can be modeled by BD sequences with R=127 and S=125. Finally, we’ll touch on generalizations like the Lamech–Moser theorem and other constants that yield complementary BD sequences, revealing the broader structure behind this elegant arithmetic.

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