CN-SMO 2008 Problem 8

Let $n$ be a positive integer. $f(n)$ denotes the number of $n$ -digit numbers $\overline{a_1a_2\cdots a_n}$ (wave numbers) satisfying the following conditions : (i) for each $a_i \in\{1,2,3,4\}$ , $a_i \not= a_{i+1}$ , $i=1,2,\cdots$ ; (ii) for $n\ge 3$ , $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$ . (1) Find the value of $f(10)$ ; (2) Determine the remainder of $f(2008)$ upon division by $13$ .