I have a very interesting number theory problem. Let $ S_n $ be a number consisting of only $n$ ones. For instance, $S_1=1\\S_2=11 \\ S_4=1111$ The problem is to prove that the sum of the digits of $S^2_n$ can be calculated using the formula $81\cdot \left( \left\lfloor \frac{n}{9} \right\rfloor + \left( \frac{n}{9} - \left\lfloor \frac{n}{9} \right\rfloor \right)^2 \right)$
I would be very grateful for help. I don't even know how to start...