Brazilian Math Olympiad

Let $S_n$ be the $n$-th Somanacci number. Then, for $n \ge 5$, $S_n = S_{n-1} + S_{n-2} + \dots + S_1 = S_{n-1} + S_{n-1} = 2S_{n-1}$, because $S_{n-1} = S_{n-2} + \dots + S_1$. So, since $S_4 = S_3 + S_2 + S_1$, $S_n = (S_1 + S_2 + S_3) \cdot 2^{n-4}$. Since $2012 = 2 \cdot 1006 = 2^2 \cdot 503$, the answer is the number of ways of writing $1006$ as sum of three positive integer numbers plus the number of ways of writing $503$ as sum of three positive integer numbers, that is $\binom{1005}{2} + \binom{502}{2} = 630261$.