The integers integers are put on the sides and vertices of an equilateral triangle. Any integer put on a vertex of the triangle will count for both sides intersecting at that vertex. If all three vertices have an integer fixed on them and the sum of the integers on each side (including the vertices) is , find number of distinct triangles that can be formed. Any triangle that can be formed as a rotation of another triangle is not considered distinct. Order of the integers on the sides is irrelevant as well.