HMMT 2009 U31

How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$ ? (That is, how many lines pass through three of the lattice points $\left(a_{1}, \ldots, a_{n}\right)$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\} ?$ ) Express your answer in terms of $n$.