Can you explain the pattern?

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[razaele]

Looks like Fibonacci without the second 1... The n=9 should be 76

[quay]

Is that Lucas' number but skipping 0th term, which was 2?

[jaj_lags]

I came up with a neat proof of why we have the same recursion rule as for the Fibonacci / Lucas sequence.

First, let's think of each n-grid tiling as a sequence of "cuts" and "non-cuts" around a circle, starting from a fixed angle. Since we can only cover at most two sectors, a "non-cut" must always follow, and be followed by, a "cut".

With this idea, we can describe an n-grid tiling as a binary string of n bits, where '0' indicates a non-cut and '1' indicates a cut.
Importantly, no two '0's can be "circularly consecutive", i.e. either consecutive or located in the first and last position respectively.
(In the degenerate case n = 1 these two positions coincide, thus we assume '1' to be the only acceptable string.)

For instance, '01101' is a valid string, which means "cut on the second, third and fifth angle";
the strings "10010" and "01010" are not valid.

Let's call the set of such strings S(n).
We can create a map from the (disjoint) union of S(n-1) and S(n-2) to S(n), in the following way:
- Given s in S(n-1), we map it to s + '1' ;
(+ denotes string concatenation.)
- Given s in S(n-2):
- If the first digit is 0, we map it to s + '01' ;
- If the first digit is 1, we map it to s + '10' .

This map is a 1-1 correspondence, which explains why the aforementioned pattern holds.

[Kreypossukr]

Huh ? I didn’t even understand the rules

[darthTwin]

Very interesting, and that music hits, surprisingly

[leznikm]

Let the number of tilings of an n grid be f(n). Consider the first tile placed
If it covers 1 cell then there are n-1 remaining cells. These can be tiled in f(n-1) ways.
If the first tile covers 2 cells, then there are n-2 remaining cells. These can be tiled in f(n-2) ways.
Therefore f(n) = f(n-1) + f(n-2) where f(1) = 1 and f(2) = 3. This is a similar sequence to the Fibonacci sequence where the two starting terms are 1 and 1.
These numbers are called the Lucas numbers

[ChessGamer]

Fibonacci sequence but it starts with 1 and 3 instead

[jakobr_]

The fact that different rotations of the same arrangement count differently is throwing me off

[fanamatakecick]

This is incoherent as far as I’m concerned

[GraydenMarvelyou]

Add the two biggest number to get the next number

[HJ_]

The fibonacci sequence starting with 1, 3 rather than the regular 1, 1 or 0, 1 starting numbers

[ymsyms]

Each number equals the sum of the 2 preceding numbers

[cam-inf-w]

But how is 1, 3, 4 exponential?....wait....it just adds the number before it? Hm

[yellowonpurple]

I understand the Lucas numbers part, but what governs the pattern itself?

[SkyBlue-fm]

No. Of combinations the hole/fraction will get

[nebxonexicron]

Sing sing red indigo

See if you get that reference

[devonharvey]

Its how many times its change form or tile placement or whatever tf that’s called, but it’s the number of which the amount of transformations it undergoes(but I think I’m very wrong) someone teach me the concept that’s in this video

[jamiepayton]

Really didn’t follow what the colours were doing here, they made it more confusing.