Good day! I’m currently investigating how different shapes can be made from a particular number of squares. I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation? Researching and manually listing the possible shapes that can be formed, my observations are the following:
Without flip and rotation:
Number of squares -Number of shapes made
1- 1
2- 1
3- 2
4- 5
5- 12
6- 35
7- 108
8- 384 edit: should be 369
With flip and rotation:
Number of squares- Number of shapes made
1- 1
2- 2
3- 6
4- 19
5- 63
6- 208 edit: should be 216
Any help you might extend will be highly appreciated.
PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this.
Many thanks!
Edit: Added a picture because I don't think I was able to point my message clearly :D