I encountered this fact yesterday: $1$ and $4900$ are the only squares as the sum of $1+4+9+\ldots +n^2$. I was trying to solve this problem using my knowledge of elementary number theory. I reduce it to the point:
Show that $(a,b,c)=(2,5,7)$ is the only positive integer solution to $$ \left\lbrace \begin{array}{} 6\times a^2+1=b^2 \\ 12\times a^2+1=c^2 \end{array} \right. $$ (then let $n=6\times a^2$, you get $4900= 24\times25\times49/6$)
I recognize these as Pell's equations, but I don't know how to proceed.
You are exploring what has been called the Cannonball problem. Édouard Lucas formulated the cannonball problem as a Diophantine equation $$\sum_{n = 1}^N n^2 = M^2$$ or $$\frac{1}{6} N(N + 1)(2N + 1) = \frac{2N^3 + 3N^2 + N}{6} = M^2.$$
Lucas conjectured that the only solutions are $N=1, M=1$, and $N=24, M=70$, using either $1$ or $4900$ cannon balls (as it seems you have observed). It was not until 1918 that G. N. Watson proved it, using a method that is far beyond on the scope of elementary number theory. However it was proved in $1990$ in just over $4$ pages, if you are interested in seeing a simpler proof.