OEIS A000093: Floors, Circles, and the Hidden Geometry of Squares

In this episode we dive into A000093, the floor of N^(3/2). We’ll see how this deceptively simple formula links to a web of ideas: the interior lattice points of a circle (A077121 with the “minus 1” adjustment to exclude points on the circumference), the square-root–based family that starts with floor(sqrt(N)) (A000196) and extends to floor((sqrt(N))^3), floor((sqrt(N))^5), and beyond, revealing the distribution of squares and the gaps between them. We’ll explore the broader connections to related sequences such as A000578 (counting squares up to n), A074704 (sum of the first n squares), and A0002821 (ways to represent n as a sum of two squares), and discuss how a simple floor operation encodes deep number-theoretic structure and invites questions about higher powers and square-distribution patterns.

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