Case 1:  There are 1 triominoes numbered to 1.
Case 2:  There are 4 triominoes numbered to 2.
Case 3:  There are 11 triominoes numbered to 3.
Case 4:  There are 24 triominoes numbered to 4.
Case 5:  There are 45 triominoes numbered to 5.
Case 6:  There are 16909 triominoes numbered to 37.
Case 7:  There are 268181 triominoes numbered to 93.
Case 8:  There are 333400 triominoes numbered to 100.