Calculate the number of free n-polyominoes

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There is currently no known formula to calculate the number of free n-polyominoes, only bounds exist. The problem remains unsolved, and it is unclear whether it has been proven to be insoluble. Although it seems straightforward to derive a formula, the complexity of the problem suggests otherwise. Historical references, such as those by Martin Gardner, indicate that this issue has been recognized for decades. Overall, the challenge of finding a solution continues to intrigue mathematicians.
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As far as I can make out, there is no formula available to calculate the number of free n-polyominoes, only bounds. Can you please confirm whether this is the case. If there is a formula, could you please point me to it. If there is not, is the problem just unsolved or has it been shown in some way to be insoluble?

At first sight, it looks as if it ought to be a simple formula to find, so if one cannot be found, is it possible to identify the reason why?

Thanks in anticipation.
 
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I think Martin Gardner has written in one of his many books that the problem is unsolved. That was a while ago though, back in the 60ies/70ies and a lot has happened since then. Typically though, most problems of this sort are difficult to solve.
 
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