In this problem, we consider the set of all the binary matrices of size `2\times n`, denoted by `\mathcal B_{2,n}`. Suppose `M\in\mathcal B_{2,n}` and `P\in \mathcal B_{2,2}`, we define `mch(P,M)` as the number of occurences of `P` as blocks in `M`. If `mch(P,M)=0`, we say `M` avoids `P`.
For example, `P=[[0,1],[1,1]]\in \mathcal B_{2,2}` and `M=[[0,0,1,1,0,1,0,1],[1,1,1,0,1,1,1,0]]\in \mathcal B_{2,8}`.
We see there are two occurences of `P`: at the second column and fifth column and hence `mch(P,M)=2`.
`a(n)` is the number of elements in `\mathcal B_{2,n}` avoiding `P=[[0,1],[1,1]]`.
For example, `a(1)=4`, `a(2)=15`.
Find `a(n)`.