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)`.