OEIS A000194: Nearest integer to square root of n

We explore A000194, the sequence that maps n to the nearest integer to sqrt(n). We explain why 0 appears once and each k ≥ 1 appears 2k times, derive the rounding window k−1/2 < sqrt(n) < k+1/2, and translate that into an integer range for n. We look at alternative definitions: the oblong-root ceiling, floor-based formulas, and the intuitive floor((sqrt(n)+0.5)). We connect the pattern to oblong numbers, interval endpoints linked to A0002061 and A0002378, and discuss deeper links to Pell equations and Romanujan theta functions via generating functions. It’s a compact tour of how a simple rounding rule reveals rich number-theoretic structure.

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