The best polynomial bounds for the number of triangles in a simple arrangement of n pseudo-lines

It is well-known and easy to observe that affine (respectively projective) simple arrangement of n pseudo-lines may have at most n(n − 2)/3 (respectively n(n − 1)/3) triangles. However, these bounds are reached for only some values of n (mod 6). We provide the best polynomial bound for the affine and the projective case, and for each value of n (mod 6).