OEIS A000249: Nearest integer to the modified Bessel function K_n(5)

We peel back the math behind A000249, defined as the nearest integer to K_n(5), the modified Bessel function of the second kind. We introduce K_alpha(x), contrast it with the ordinary Bessel functions, and explain why K behaves exponentially for real x while diverging at zero. We cover integral representations, key asymptotics, and what they imply for the behavior as the order n grows with fixed x = 5 (and thus how the sequence tends to small values when rounded). Along the way we connect to the physics and geometry that make Bessel functions pop in cylindrical problems, diffusion, and more, and clarify how understanding K_n(x) sheds light on this OEIS entry without computing term by term.

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