Polyominoes have been obsessing me. Not tiling but counting them. No formula has been yet found. Their free numbers (i.e up to rotations and flipping) go

  1, 1, 2, 5, 12, 35, 108, 369, 1285, ...

Meaning for ex. there are 5 tetrominoes :

  ■
  ■        
  ■        
  ■ 

  ■        
  ■        
  ■ ■ 

  ■        
  ■ ■       
  ■ 

  ■        
  ■ ■       
    ■ 

  ■ ■        
  ■ ■       

The myterious sequence follows a ~4 growth rate.

I've been (nowhere as a mathematician) exploring this problem for years, generating them and trying in vain to find patterns in their properties.

I'd love readers with a high view in combinatorics telling what they think of this problem. Do you think a formula will one day be found or rather that there can't be a closed one for some necessary reason ?