Find a generating function involving a tiling problem. Let hn be the number of ways to tile a 1 times n rectangle with 1 times 1 tiles that are red or blue and 1 times 2 tiles that are green, yellow, or white. Find a closed formula for H(x)=sum_{n >= 0} h_n x^n

Francisco Proctor 2022-07-18 Answered
Find a generating function involving a tiling problem
Let h n be the number of ways to tile a 1 × n rectangle with 1 × 1 tiles that are red or blue and 1 × 2 tiles that are green, yellow, or white. Find a closed formula for H ( x ) = n 0 h n x n .
I am unsure how to start this problem. Is there a way to solve this without using recurrence relations?
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Answers (2)

penangrl
Answered 2022-07-19 Author has 17 answers
Step 1
A recurrence relation is a good place to start. By considering the possibilities for the first tile, we obtain h n = 2 h n 1 + 3 h n 2 for n 2. The boundary conditions are h 0 = 1 and h 1 = 2.
Step 2
Now the recurrence relation and boundary conditions yield H ( x ) 1 2 x = 2 x ( H ( x ) 1 ) + 3 x 2 H ( x ) , from which we find that H ( x ) = 1 1 2 x 3 x 2 ..
Partial fraction decomposition then yields H ( x ) = 1 / 4 1 + x + 3 / 4 1 3 x ,, which immediately implies the explicit formula
h n = 1 4 ( 1 ) n + 3 4 3 n = ( 1 ) n + 3 n + 1 4 .
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Marcelo Mullins
Answered 2022-07-20 Author has 1 answers
Step 1
In general, if you want the generating function for the number of sequences of objects with a total "weight" of n, where each entry is taken independently from a set of a 1 objects of weight one, and a 2 objects of weight two, etc, then the answer is
1 1 ( a 1 x + a 2 x 2 + ) = 1 + ( a 1 x + a 2 x 2 + ) sequences of one object + ( a 1 x + a 2 x 2 + ) 2 sequences of two objects +
Step 2
In this case, you are counting sequences of tiles, where the weight of a tile is its area, and you want the generating function for tilings of weight n. There are two tiles with weight 1 (two squares), and three tiles of weight 2, so the answer is 1 / ( 1 ( 2 x + 3 x 2 ) )
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