Polyominoes (2019)
researchgate.net
researchgate.net
If you like playing with polyominoes, there is a really fun chesslike table game inspired by them called Blokus, and someone wrote an open-source engine for it: https://pentobi.sourceforge.io/
Of course, as already noted in another comment, there's the ultimate tetromino game: Tetris (literally defined as tetromino tennis)...
Uwe Rosenberg in particular has designed a bunch of games playing around with the polyomino concept, starting with Patchwork, then a trilogy of games Indian Summer, Cottage Garden, and Spring Meadow, and then culminating in a rules-heavy game A Feast for Odin.
Phil Walker-Harding designed a very popular one called Barenpark, where you cover spots on a board which unlock different types of tiles and new boards.
A recently released game that's been well received that utilizes a lazy susan for tile selection is called Planet Unknown.
There's also a tile drafting game called Isle of Cats that's quite popular as well.
A game that combines polyominoes with roll and write games that's quite popular is Cartographers.
Patchwork kicked off this new trend in board games, and was released in 2014.
1, 1, 2, 5, 12, 35, 108, 369, 1285, ...
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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 ?
The growth rate is indeed around 4 and now known to be strictly above 4: https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf
If I remember correctly, there was a non-rigorous argument for a concrete conjectured value not far above 4.
we obtain 4.00253176 as a certified lower bound
Maybe it has to do with symmetries. The proportion of symmetric polys in successive generations goes towards zero. I suspect that this 'pollutes' the asymptot.
About published research, I just glanced at the little I found, (as I often can't do more than glance, being limited in Maths.)
Also I prefer to go at it naively first.
I suspect many other folks realized this already but it was a charming lightbulb moment for myself.