\sum_{i=1}^{n-1} i^2 = \sum_{i=1}^{n-1} (n-i)^2 = \sum_{i=1}^{n-1} n^2 + \sum_{i=1}^{n-1} i^2 - \sum_{i=1}^{n-1} 2i = n^2(n-1) - n(n-1) + \sum_{i=1}^{n-1} i^2