This is getting no attention on stackexchange.

Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$.

It had escaped my attention until last week, when I wrote this answer, that the number of sequences of distinct elements of a set of size $n$ (including those of length $0$) is the nearest integer to $n!e$, provided $n\ge2$. The sequence whose $n$th term is the nearest integer to $n!e$ satifies the recurrence $x_{n+1} = (n+1)x_n + 1$.

**How widespread is this operation of mulitplying by an irrational number and then rounding, in combinatorial problems? Are there other standard examples? Is there some general theory accounting for this? And, while I'm at it, where is this in "the literature"?**

(I'm not sure whether we should include things like Fibonacci numbers or solutions of Pell's equation as examples of the same thing.)

proofthat the irrational number in question is indeed irrational. Irrationality proofs, especially of transcendental numbers, tend to be scarce, so your phenomenon tends to be scarce. Your uncertainty about quadratic irrationals is related, I think, to the fact that the proof of their irrationality is "too easy." If you allow algebraic irrationals, then I think you'll get more examples. – Timothy Chow Sep 18 '13 at 2:26