Problem 679

FREEFAREA

Let $S$ be the set consisting of the four letters
${\texttt{A'},\texttt{E’},\texttt{F'},\texttt{R’}}$.
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.

Let $f(n)$ be the number of words in $S^*(n)$ that contains all four keywords exactly once.

This first happens for $n=9$, and indeed there is a unique 9 lettered word that contain each of the keywords once: $\texttt{FREEFAREA}$
So, $f(9)=1$.

You are also given that $f(15)=72863$.

Find $f(30)$.

FREEFAREA

记$S$为包含四个字母的集合${\texttt{A'},\texttt{E’},\texttt{F'},\texttt{R’}}$。
对于$n\ge 0$，记$S^*(n)$为仅包含$S$中的字母且长度为$n$的字符串所组成的集合。
我们选择$\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$作为关键词。

记$f(n)$为$S^*(n)$中恰好包含四个关键词各一次的字符串数目。

在$n=9$时首次出现满足要求的字符串，且唯一的拥有$9$个字母的此类字符串是$\texttt{FREEFAREA}$。
因此$f(9)=1$。

已知$f(15)=72863$。

求$f(30)$。