Q:

Multiplying by irrational numbers in combinatorial problems

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 sequences 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.)

A:

This is related to Noam Elkies's answer but is not exactly the same.
Rayleigh's theorem, a.k.a. Beatty's theorem, says that if $a$ and $b$ are positive irrational numbers such that $1/a + 1/b=1$, then the sets $\lbrace \lfloor na\rfloor : n\in \mathbb{N}\rbrace$ and $\lbrace \lfloor nb\rfloor : n\in \mathbb{N}\rbrace$ comprise a partition of $\mathbb N$ into two disjoint sets.  There are connections between this theorem and various combinatorial topics, such as Wythoff's game as Noam mentioned, and combinatorics on words (Sturmian sequences).

A:

A result due to W. H. Mills (1950) mentioned in Apostol's Introduction to Analytic Number Theory (in the Historical Introduction section) states: there is a positive number $A$, which is not an integer, such that $\left\lfloor{A^{3^x}}\right\rfloor$ is a prime for all $x = 1,2,3,\ldots$. (It is probably not known if $A$ has to be irrational, but I thought this result may be of interest.)
The original reference is Bull. Amer. Math. Soc. Volume 53, Number 6 (1947), 604 (Mathematical Reviews number (MathSciNet): MR0020593) and an errata Bull. Amer. Math. Soc. Volume 53, Number 12 (1947), 1196. According to the paper $A = \lim_{n\rightarrow \infty} P_n^{3^{-n}}$, where $P_n$ is the $n$'th prime. The 1-page paper is available on Project Euclid: http://projecteuclid.org/euclid.bams/1183510803.

A:

The mutual Zugzwangs in
Wythoff's game
are
$(\lfloor \phi k \rfloor, \lfloor \phi^2 k \rfloor)$
and 
$(\lfloor \phi^2 k \rfloor, \lfloor \phi k \rfloor)$
where $k=0,1,2,\ldots$ and $\phi$ is the golden ratio $(1 + \sqrt 5)/2$ .

