S_n^2 = \frac{\left(n+1\right)^3 - 3S_n^1 - S_n^0 - 1}{3} = \frac{n^3 + 3n^2 + 3n + 1 - \frac{3n\left(n+1\right)}{2} - n - 1}{3} = \frac{2n^3 + 3n^2 + n}{6} = \frac{n\left(n+1\right)\left(2n+1\right)}{6}