What happens when you count sums of square roots rather than sums of integers? OEIS A000333 counts the number of ordered multisets L = (l1 ≤ l2 ≤ … ≤ lk) of positive integers with sqrt(l1) + sqrt(l2) + … + sqrt(lk) ≤ n. For example, A(3) = 15. The problem arose in a 1951 statistical mechanics paper by Agarwala and Alok, where distributing energy quanta over non-integer energy levels led to these non-integral partitions; Neil Sloan later cataloged the sequence, highlighting its rapid growth (1, 5, 15, 40, 98, …) and the lack of a simple generating function, alongside intriguing asymptotic structure bridging number theory and physics.

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