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As we all know, Polyominoes are shapes which consist of certain number of squares connected together. A famous videogame - Tetris - has a gameplay based around tetraminoes - polyominoes with 4 squares in them. There are 7 one-sided tetraminoes. One-sided basically means that you can't get one of them by rotating another (not reflecting).

Wikipedia page tells the number of one sided polyominoes for different number of squares in them (let's call it N). There is only 1 Monomino and 1 Domino. There are 2 Trominoes, 7 Tetraminoes, 18 Pentaminoes, 60 hexominoes and so on.

So, is there a formula which will let you find the number of one sided polyominoes for N squares? Will I be able to find, for example, how many one-sided polyominoes with 100 squares are there? With 1000 squares?

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  • $\begingroup$ A formula might be too much to ask for, but there might be some asymptotics available... $\endgroup$
    – André Henriques
    Commented Dec 20, 2021 at 12:22
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    $\begingroup$ The Wikipedia page you link gives a pretty complete answer to this. "No formula has been found except for special classes of polyominoes" and the special classes are discussed in the subsection with that name. Asymptotically, there are approximately four times as many one-sided polyominos as fixed ones. $\endgroup$
    – Peter Taylor
    Commented Dec 20, 2021 at 12:36
  • $\begingroup$ Oh my bad, did someone try to prove that it is impossible to find such formula then? $\endgroup$
    – The Fox
    Commented Dec 20, 2021 at 12:46

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See OEIS A000988 for a wealth of information on the number of one-sided polyominoes. Although there is no known formula, the growth rate of the sequence is quite regular:

OEIS graph

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