The answer is 24.

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The only two values of $n$ for which the indicated series has a Sum which is a perfect square are: 1 and 24.

If $n=1$ the Sum is $\boxed{1}$ and if $n=24$ the Sum is $\boxed{4900}$ I have checked for $1\leq n \leq 1,000,000$ while killing time in the hospital using just my TI 84+ calculator.

So, the largest value of $n$ is $\boxed{\boxed{24}}$

Also, you can check out A000330 on OEIS for further information.

A more general situation with proofs is discussed here .