[seqfan] Re: L-reptiles
qntmpkt at yahoo.com
Tue Mar 30 21:48:15 CEST 2010
thx,...possible tiling application to an algebraic set of operations, re: A004737:
1, 2, 1
1, 2, 3, 2, 1
...with row sums = (1, 4, 9, 16,...)
Given the triangle rows: (1; 1,2,1; 1,2,3,2,1;...) as polcoeff with offset 0:
q = (1 + 2x + x^2), r = (1 + 2x + 3x^2 + 2x^3 +x^4),...etc; then
(1 + 2x + 3x^2 + ...) = q(x) * q(x^2) * q(x^4) * q(^8) * ...
..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...
..................... = s(x) * s(x^4) * s(x^16)* s(x^64) * ..
From: Tanya Khovanova <mathoflove-seqfan at yahoo.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Tue, March 30, 2010 8:11:01 AM
Subject: [seqfan] L-reptiles
Carolyn Yackel gave a talk at G4G9 on L-reptiling: tiling of letter L with the same shape. (Letter L means a 2 by 2 square with one of the squares removed).
I we want to have identical tiling pieces then the shape can be tiled with 4, 9, 16, pieces:
Is it true that any square number will work?
Now suppose we do not require them to be the same size. Then if we divide the L-shape into 4 pieces, we then can divide only one of them into 4 pieces. Continuing that we get 1, 4, 7, 10 as the number of possible pieces. If we combine that with dividing some pieces into 9 smaller pieces, we can get 1, 4, 7, 9, 10, 12, 13 as the number of possible pieces.
Carolyn presented a ceramic L-shaped thingy tiled into L-shapes.
Is the number of possible pieces a good sequence to submit? Is it true that the compliment of the above sequence are impossible to achieve?
Is L-shape interesting enough to submit the sequence?
For example, an isosceles right triangle can be tiled into two smaller triangles, and hence, into any number of similar triangles if we do not require the size to be the same.
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