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 $9\le n\le 18$.