This problem is about that parking functions avoiding some consecutive patterns.
A parking function of length $n$ is a sequence $(c_1,c_2,\cdots,c_n)$ of positive integers such that there exists a permutation $\sigma\in S_n$ such that $c_{\sigma(i)}\le i$ for any $1\le i \le n$.
For example, there are $3$ parking functions of length $2$, namely $(1,1)$, $(1,2)$ and $(2,1)$. The number of parking functions of length $n$ is $(n+1)^{n-1}$.
$a(n)$ is the number of parking functions of length $n$ such that there are no consecutive repeats. For example, $a(2)=2$ becasue there are only two such parking functions, $(1,2)$ and $(2,1)$.
$b(n)$ is the number of parking functions of length $n$ such that the difference between an integer and its previous integer is not $1$. For example, $a(2)=2$ becasue there are only two such parking functions, $(1,1)$ and $(2,1)$.
$c(n)$ is the number of parking functions of length $n$ such that the absolute value of difference between an integer and its previous integer is not $1$. For example, $a(2)=1$ becasue there is only one such parking function, $(1,1)$.
More generally, we shall consider an arbitrary pattern.
Find $a(n)$, $b(n)$ and $c(n)$.