OEIS A000209: Nearest integer to tan

An accessible exploration of A000209, the sequence formed by taking the nearest integer to tan(n) with n in radians. We unpack how tan behaves on the unit circle, why its values spike near odd multiples of pi/2, and how integers n sample the real line densely modulo pi. We connect these dynamics to number theory ideas like Diophantine approximation and the influence of pi's irrationality (and transcendence) on the sequence’s wild jumps, while also anchoring the discussion with practical geometric intuition from trigonometry.

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