OEIS A000196: Integer part of the square root

We dive into A000196, the floor of sqrt(n). From counting squares ≤ n to the recurring pattern that each value k appears 2k+1 times, this humble sequence links to divisor geometry, alternate bases based on squares, and its role as a left inverse of squaring. Along the way we’ll explore generating functions, theta functions, and the web of cross-references that reveal how a simple rule opens doors to deep number-theoretic connections.

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