Due to the assignment of $\left(a_{i}, b_{i}\right)$ being 4 cases: $(0,0),(0,1),(1,0),(1,1)$, we can consider $\left(a_{i}, b_{i}\right)$ as a letter, denoted as $(0,0),(0,1),(1,0),(1,1)$ being $A, B, C, D$ respectively. Thus, the interesting array $T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right)$ can be seen as a permutation of $n$ letters composed of $A, B, C, D$.
According to the problem, when $\left(a_{p}, b_{p}\right)=(1,0)$, the subsequent $b_{q} \neq 1$, meaning $C$ cannot be followed by $B, D$; when $\left(a_{q}, b_{q}\right)=(1,0)$, the preceding $a_{p} \neq 0$, meaning $C$ cannot be preceded by $A, B$.
Therefore, the permutation of the interesting array containing the letter $C$ can only be $\underbrace{D D \cdots D}_{x \uparrow} \underbrace{C C \cdots C}_{y \uparrow} \underbrace{A A \cdots A}_{z \uparrow}$, where $x+y+z=n, x, z \geqslant 0, y \geqslant 1$.
Let $x^{\prime}=x+1, y^{\prime}=y, z^{\prime}=z+1, x^{\prime}+y^{\prime}+z^{\prime}=n+2$, the number of positive integer solutions is $C_{n+1}^{2}=\frac{n(n+1)}{2}$, so the number of interesting arrays in this case is $\frac{n(n+1)}{2}$;
When the interesting array does not contain the letter $C$, any permutation of $A, B, D$ is valid, so the number of interesting arrays in this case is $3^{n}$. In summary, the number of interesting arrays is $3^{n}+\frac{n(n+1)}{2}$.
11. Given an integer $n(n \geqslant 2)$. For a $2 n$-tuple ordered array $T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right)$, if each component of $T$ is 0 or 1, and for any $p, q(1 \leqslant p<q \leqslant n)$, we have $\left(a_{p}, b_{p}, b_{q}\right) \neq (1,0,1),\left(a_{q}, b_{q}, a_{p}\right) \neq(1,0,0)$, then $T$ is called a "fun array". Find the number of fun arrays.