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How to find perfect squares such that their base 9 representation is all 1s?

I am working on solving all of the problems in from the Macalester College Problems of the Week that are available in [this](https://stanwagon.com/potw/) archive, and am currently working on MacPOW 1114:
>### Squares of 1
>
>Find all perfect squares whose base 9 representation consists of only 1s.

Here is my current progress on this:
* First of all, we can see that 1 is the only trivial solution - there of course have to be other non-trivial solutions.

So really our problem here is finding square numbers of the form$$\sum_{n=0}^k9^n$$however I'm not really sure about how I should go about finding square numbers of that form. I guess that if I replace 9 with $x$, then the problem really is equivalent to finding polynomials $p(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+1$ such that$$p(x)^2=x^{2n}+x^{2n-1}+x^{2n-2}+\cdots+1$$although my question is: How do I solve this problem? I'm not sure what to do from here, so any help is appreciated.