OEIS A000329: Tangent Iteration Sequence

We explore A000329, the tangent-iteration sequence defined by b(0) = 1 and the nearest integer to b(n), where b(n) = tan(b(n-1)). The interplay between the continuous, blow-up behavior of tan near odd multiples of π/2 and the discrete rounding step yields a surprisingly erratic sequence (with terms wandering through 1, 2, 75, -1, -1, -2 … and beyond). We unpack how the sensitivity of tan to its input, combined with rounding, acts as a powerful nonlinearity that can send the next term in a completely different direction from tiny fluctuations. This makes numerical computation extremely delicate: standard floating-point is far from enough, and interval arithmetic or very high-precision arithmetic are used to bound errors and verify terms. Some computations reportedly require tens of thousands of bits of precision to stay on the true trajectory, illustrating the fragile, chaotic-like dynamics of a simple rule. We’ll also discuss how small changes in the starting value could dramatically alter the long-term behavior and what this reveals about deterministic maps in number theory and numerical analysis.

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