We call an integer sided triangle $n$-pandigital if it contains one angle of $120$ degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.
For example, the triangle $(217, 248, 403)$ is $9$-pandigital because it contains one angle of $120$ degrees and the sides written in base $9$ are $261_9, 305_9, 487_9$ using each of the $9$ digits of that base once.
Find the sum of the largest sides of all $n$-pandigital triangles with $9 \le n \le 18$.
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