OEIS A000213: The Floor of n^2/3 and Its Surprising Connections

A deceptively simple sequence AN = floor(n^2/3) with offset 0 unlocks a surprising web of connections across algebra, geometry, and combinatorics. We trace how it appears as the determinant of a structured matrix, bounds for two-generator directed Cayley graphs of diameter N−2, a maximal rectangle-packing bound for 1×3 tiles in n×n squares, and extremal graphs without K4, along with a variety of combinatorial interpretations—from partitions of 2n into three parts to unimodal triples summing to n+1, and even ties to elliptic curves. Along the way we encounter the work of Cloyder, Steffen, Kamenevsky, Yang, Bala, Barker, Zakharov, Kirchner, Wiseman, and others, illustrating how a simple floor function reveals a rich, interconnected mathematical landscape.

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