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This Python program works by counting the triangles. It runs in 37ms.

Solution:The triangles have sides of length 3, 4, 6 and 7.There is only one way to to pick the triangles for the case of ET[7] = 118.

The next case happens at ET[10] = 315, and there are three sets of triangles that can be chosen:

ET[10] = ET[4] + ET[7] + ET[8]

ET[10] = ET[2] + ET[4] + ET[5] + ET[9]

ET[10] = ET[1] + ET[2] + ET[3] + ET[5] + ET[6] + ET[8]

The sequence ET[n] is A002717 in OEIS. [ http://oeis.org/A002717 ]

So we could just use the following function to compute ET[n]: