1993 Putnam A6

The infinite sequence of 2's and 3's

$$

\begin{aligned}

&2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\

&3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\dots

\end{aligned}

$$

has the property that, if one forms a second sequence that records the

number of 3's between successive 2's, the result is identical to the

given sequence. Show that there exists a real number $r$ such that, for

any $n$, the $n$th term of the sequence is 2 if and only if $n = 1 +

\lfloor rm \rfloor$for some nonnegative integer$m$. (Note:$\lfloor x

\rfloor$denotes the largest integer less than or equal to$x$.)

2 和 3 的无限序列

$$

\开始{对齐}

&2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\

&3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\点

\结束{对齐}

$$

具有这样的性质：如果形成第二个序列来记录

连续 2 之间 3 的数量，结果与

给定的序列。证明存在一个实数$r$，对于

任何 $n$，序列的第 $n$ 项为 2 当且仅当 $n = 1 +

\lfloor rm \rfloor$表示某个非负整数$m$。 (注：$\l地板x

\rfloor$表示小于或等于$x$ 的最大整数。)