Looking at this question, I misread "dodecagon" as "dodecahedron". I think the latter is a cool problem, so I'm posing it as a question of its own :)
Starting from one vertex of a dodecahedron, an ant wants to reach the opposite vertex of the dodecahedron, moving to adjacent vertices. If $p_n$ is the number of such paths with length $n$, compute $p_1+p_2+\dots+p_{12}$.