How many unique integer solutions to the circle $x^2+y^2=2n^2$?

Question

I am examining the equation of this circle and what to find the number of integer solutions,

$x^2+y^2=2n^2$

but with a few constraints,

I'd like the number of pairs of unique integer solutions to $x^2+y^2=2n^2$ such that,

$x \neq y \neq n$ and $x,y,n > 0$

In addition when I say unique solution I mean no double counting so, (1,10) (5,5) (10,1) would have 3 unique integer solutions.

It might help to note that $2n^2$ is even.

Some background, this is my first exposure to this topic. I watched the 3b1b video on it and I find the geometric intuition very cool. I understand that this is the equivalent to asking the number of lattice points hit by the circle with the constraints above. Here are some things that I would like to know:

~~Is there a limit to how many integer solutions with the constraints above (a maximum number)?~~
If not, is there a way to search for values of n with larger amounts of unique solutions?
Are there any interesting or unique things you noticed while looking at this specific circle?

Once again, this is my first exposure to the topic so please forgive me if some of these questions are trivial or unanswerable. I have tried to find some sort of pattern or regularity for this but haven't had any luck (I assume this might be because the problem is related to prime factorization)

Also, I have seen the post on the generalized solution to this (I'll admit I don't fully understand it yet) but I am asking in regard to this specific example as well as these special constraints.

Edit: @automaticallyGenerated has found this sequence. Which shows the question is basically asking about the number of pythagorean triples containing a fixed even integer.

Edit 2: Since posting I have found that the number of solutions generally increases indefinitely as $n \rightarrow \infty$. Wolfram alpha can tell you very quickly the integer solutions to this equation. I would still like to know if there is a mathematical way to find the number of triples "easily" for a given n.

Only the numbers in this OEIS sequence will have a nonzero number of pairs. — Varun Vejalla, Oct 04 '19 at 22:00
Actually, the formula for this can be found in this OEIS sequence. — Varun Vejalla, Oct 04 '19 at 22:03
Thanks, I knew that was a formulation of the question but I did not know that there was a sequence for it! — PMaynard, Oct 04 '19 at 22:06
If you meant $n$ is fixed, searching for solutions in $x,y$, this problem is well-know and the solution is from the Gaussian primes : distinguishing between prime $p \equiv 3\bmod 4$, showing $|x+iy|^2= p$ has no solution, and $p \not \equiv 3\bmod 4$, showing $|x+iy|^2=p$ has a solution. Factorizing $2n^2$ and multiplying the solutions for each prime factor we get all the solutions for $|x+iy|^2=2n^2$. — reuns, Oct 05 '19 at 22:34

How many unique integer solutions to the circle $x^2+y^2=2n^2$?

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