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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.

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  • $\begingroup$ How about flipping the shape over? $\endgroup$ – MaxW Apr 11 '16 at 22:42
  • $\begingroup$ @MaxW, no flipping, so you can have mirror images that would count as two shapes. $\endgroup$ – Nathan Dyck Apr 11 '16 at 23:08
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    $\begingroup$ 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 $\endgroup$ – Blue Apr 11 '16 at 23:34
  • $\begingroup$ thanks @Blue, that'll do it. $\endgroup$ – Nathan Dyck Apr 12 '16 at 0:00
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The comment by Blue already gave some useful links: these shapes made out of triangles are called polyiamonds, a generalization from the diamond made of two triangles. The version using squares instead of triangles would be called polyominos.

Most likely you can find the counts you're after on OEIS. But where the comment suggests A000577 which consideres reflected polyiamonds the same, the edited version of your post asks to count them as different (so you're dealing with “one-sided” polyiamonds). That would be A006534 instead, still for the case with holes. Your post doesn't exclude holes, and OEIS currently doesn't have a sequence for one-sided hole-less polyiamonds. Since holes can only occur for sizes of at least nine triangles (noniamonds and above), the distinction is irrelevant for smaller polyiamonds.

Finding a simple formula for these beasts would be quite tricky. My best bet at the moment would be pretty brute-force enumeration. From all the references mentioned on the A006534 page, Counting hexagonal and triangular polyominoes by Lunnon sounds the most promising, if you can get access to that. So far I've only read the abstract.

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