An alphabet set $\{\alpha_1, \alpha_2, \ldots, \alpha_8\}$ is used to form a string. A legal string cannot have consecutive $\{\alpha_1\alpha_1, \alpha_1\alpha_2, \alpha_2\alpha_1, \alpha_2\alpha_2\}$ in any part of the string. Suppose $P_{n-1}$ is the number of valid strings of length $n$.