Triangulating A Triangle Of Coins!

A triangles has a non-negative integer \(n\) coins on all its sides as demonstrated above for \(n=9\). For how many values of \(n<1000\) can we build such a triangle with side-length \(n\) using only the below coin triangles?


Determine the rule for \(n\) for which it is possible to tile the triangle.

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