Lucas Number Circular Tilings (synthwave enumeration)

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[PhilBoswell]

My guess about the Lucas Number link is that you're always adding a 1-tile in between two of the tiles in the (n-1) tiling, or a 2-tile in between two of the tiles in the (n-2) tiling, so you're getting some combination of the (n-1) and (n-2) counts while accommodating duplication of patterns.

[debblez]

If you consider different rotations to be equivalent, you’ll count 1, 2, 2, 3, 3, 5, 5, 8, 10, 15, and 19 distinct tiling(s)

[zengakukatsu]

You just add the previous 2 numbers. But why? No idea, kinda wanna look into it now. It works exactly like Fibonacci except it starts with a different seed. Strange that it also seems to converge to the golden ratio when you divide 2 consecutive numbers. Does it? If it does I wonder what the basis behind that is, does any seed eventually converge to the golden ratio? Never looked into the math behind that before, seems interesting. Does it follow other rules and patterns of the Fibonacci sequence, like the diagonal pattern within Pascals Triangle? Fibonacci always kinda felt like a party trick to me, but for some reason showing the generality of its properties in a non-Fibonacci sequence has sparked my interest. Great video!

[tsukyow]

Didn’t understand no shit but still looks cool