Dear All,

my following question may be known and ought to be known, so in case it is folklore please could you give me the references.

To start, it is obvious that growth of rational languages are always either polynomial or exponential. That is, if $L$ is a regular language, then the sequence $a_n$, where $a_n$ is the number of all words $w\in L$ of length $\leq n$, grows polynomially or exponentially.

But what I am really curious about is

$\textbf{Question}:$ If $L$ is a regular language, is then the correspondent growth zeta-function $\sum\limits_{n=1}^{\infty}a_nx^n$ rational?