How many different shapes can you construct with n equilateral triangles?

If you have n equilateral triangles, and you want to connect them all to each other at the edges, how many different shapes can you make? Triangles are identical in size and shapes that are rotationally congruent should be not be counted multiple times. Mirror images should be counted as different.

• How about flipping the shape over? – MaxW Apr 11 '16 at 22:42
• @MaxW, no flipping, so you can have mirror images that would count as two shapes. – Nathan Dyck Apr 11 '16 at 23:08
• These shapes are known as "polyiamonds". The number of polyiamonds with $n$ triangles is sequence A000577 at OEIS; if you don't allow holes, the sequence is A070765 – Blue Apr 11 '16 at 23:34
• thanks @Blue, that'll do it. – Nathan Dyck Apr 12 '16 at 0:00