  First, we consider order-$n$ ribbon tilings of an $M$-by-$N$ rectangle
$R_{M,N}$ where $M$ and $N$ are much larger than $n$. We prove the existence of
the growth rate $\gamma_n$ of the number of tilings and show that $\gamma_n
\leq (n-1) \ln 2$. Then, we study a rectangle $R_{M,N}$ with fixed width $M=n$,
called a strip. We derive lower and upper bounds on the growth rate $\mu_n$ for
strips as $ \ln n - 1 + o(1) \leq \mu_n \leq \ln n $. Besides, we construct a
recursive system which enables us to enumerate the order-$n$ ribbon tilings of
a strip for all $n \leq 8$ and calculate the corresponding generating
functions.
