A000167: The Nearest Integer to K_n(2) — A Bessel Whisper in Numbers

In this episode we dive into A000167, the sequence a(n) = round(K_n(2)) where K_n is the modified Bessel function of the second kind. We’ll unpack what the modified Bessel function is, why sampling at the fixed value 2 matters, and how taking the nearest integer creates a discrete staircase from a smooth, continuous curve. We’ll explore the exact recurrence K_{n+1}(2) = n K_n(2) + K_{n-1}(2), what the tail behavior looks like as n grows, and how the zeros and the overall shape of the Bessel function shape the sequence. Finally, we’ll touch on what number-theoretic properties the integers themselves exhibit and how this bridge between continuous functions and discrete sequences in the OEIS can spark further curiosity.

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