The sequence of the common sides is {5, 6, 10, 12, 20, 24, 40, 48, 51, 90, 108, 208, 384, 408, 720, 864, 918, 1620, 1944, 3880, 4656, 6240, 6336, ...}
a(n) = 6*2^n for n = 0, 1, 2,..., 7, and then this property disappears.
The area is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where the semiperimeter s = (a + b + c)/2.
The following table gives the first values (n, A, a, b, c) where a <= b <= c are the integer sides of the triangles.
+----+------+-----+-----+-----+
| n | A | a | b | c |
+----+------+-----+-----+-----+
| 0 | 6 | 3 | 4 | 5 |
| 1 | 12 | 5 | 5 | 6 |
| 2 | 24 | 6 | 8 | 10 |
| 3 | 48 | 10 | 10 | 12 |
| 4 | 96 | 12 | 16 | 20 |
| 5 | 192 | 20 | 20 | 24 |
| 6 | 384 | 24 | 32 | 40 |
| 7 | 768 | 40 | 40 | 48 |
| 8 | 1080 | 48 | 51 | 51 |
| 9 | 1080 | 51 | 51 | 90 |
| 10 | 3888 | 90 | 90 | 108 |
| 11 | 4320 | 108 | 116 | 208 |
+----+------+-----+-----+-----+
|