I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even necessary). I did a quick run through with a python script and of course this seems totally devoid of a computational pattern

I did find out that the sum of squares is the following...

$\sum^{n}_{i=1} i^2 = \frac{n(n+1)(2n+1)}{6}$

But I still can't pull the answer out...

Related: What makes positive integers so special? This is what I kept running through my head and couldn't get it. Granted, my number theory is especially poor. As far as I can tell this isn't trivial, but I could be quite wrong.

I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even necessary). I did a quick run through with a python script and of course this seems totally devoid of a computational pattern. Why is 4900 (and 1 of course) the only number such that this works?

I did find out that the sum of squares is the following...

$\sum^{n}_{i=1} i^2 = \frac{n(n+1)(2n+1)}{6}$