For a polynomial $ f$ define $ \Delta_c f$ by $ (\Delta_c f)(x) : = f(x + c) - f(x)$. This is a polynomial of degree one less than $ f$ (or the $ 0$ polynomial if $ f$ was constant). Then consider $ \Delta_4 \Delta_2 \Delta_1 f$, it is $ 0$. Expanding it gives that it is $ (x + 7)^2 - (x + 6)^2 - (x + 5)^2 + (x + 4)^2 - (x + 3)^2 + (x + 2)^2 + (x + 1)^2 - x^2 = 0$. This reduces the original problem to checking those cases with $ n \leq 7$, which can easily be done. One couls also use $ \Delta_2 \Delta_1 f$ to be $ 4$, easily reducing the problem to $ n \leq 3$, making the checking easier. This also shows that one can replace the square by arbitrary powers $ k^n$ if one just changes $ 4$ into a big enough number (the biggest value occuring in the first $ 2^{n + 1}$ numbers is the best possible bound, a more general bound is $ n! \cdot 2^{\frac {n(n - 1)}{2}}$ or $ 2^{n^2}$). See also http://www.mathlinks.ro/viewtopic.php?t=30998 for a related problem.