You can form (n-1)² 2 by 2 squares.

You can form (n-2)² 3 by 3 squares.

You can form (n-3)² 4 by 4 squares.

. . . You can form 1² n by n square.

The sum of all these sums is n*(n+1)*(2n+1)/6

Thanks to Guy de Kindler for this one.