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Question

# a) Find a recurrence relation for the number of ways to lay out a walkway with slate tiles if the tiles are red, green, or gray, so that no two red tiles are adjacent and tiles of the same color are considered indistinguishable. b) What are the initial conditions for the recurrence relation in part (a)? c) How many ways are there to lay out a path of seven tiles as described in part (a)?

Solution

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(a) Let $a_n$ represent the number of ways to lay out a walkway with $n$ slate tiles if the tiles are red, green or gray, no two red tiles are adjacent and tiles of the same color are considered indistinguishable.

$\textbf{First case }$ The right-most tile is green, then the walkway still needs $n-1$ slate tiles and there are then $a_{n-1}$ ways to tile the remaining walkway according to the specifications.

$\textbf{Second case }$ The right-most tile is gray, then the walkway still needs $n-1$ slate tiles and there are then $a_{n-1}$ ways to tile the remaining walkway according to the specifications.

$\textbf{Third case }$ The right-most tile is red and the tile next to it is green, then the walkway still needs $n-2$ slate tiles and there are then $a_{n-2}$ ways to tile the remaining walkway according to the specifications.

$\textbf{Fourth case }$ The right-most tile is red and the tile next to it is gray, then the walkway still needs $n-2$ slate tiles and there are then $a_{n-2}$ ways to tile the remaining walkway according to the specifications.

Adding the number of sequences of all three cases, we then obtain:

$a_n=a_{n-1}+a_{n-1}+a_{n-2}+a_{n-2}=2a_{n-1}+2a_{n-2}$

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