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For n>2, a(n) + (-1)^ceiling(n/2) is the number of ways to tile this strip of length n-1, with a central staircase, using unit squares and dominoes:
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For n>2, a(n) + (-1)^Ceiling(n/2) is the number of ways to tile this strip of length n-1, with a central staircase, using squares and dominoes: _
_
_______|_|_|_________
_______|_|_|_________|_|_|_|_|_|_|_|_|_, and Runhe Zhang, Sep 07 2024
For n>2, a(n) + (-1)^Ceiling(n/2) is the number of ways to tile this strip of length n-1, with a central staircase, using squares and dominoes: _
_|_|
_______|_|_|_________|_|_|_|_|_|_|_|_|_|_|_|. - Greg Dresden, Sep 07 2024
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For n >= 2, a(n) is also the number of ways to use white dominoes and black and white squares to tile this strip of length n which has a 4-cell zig-zag in the center with the rule that black squares can must appear exactly twice and can only appear in the four center zig-zag cells. Here is the strip of length 7:
|_|_|___|_|___|. - Greg Dresden, and Emma Li, Sep 06 2024
For n >= 2, a(n) is also the number of ways to use white dominoes and black and white squares to tile this strip of length n which has a 4-cell zig-zag in the center with the rule that black squares can appear exactly twice and can only appear in the four center zig-zag cells. Here is the strip of length 7:
_
_____|_|_____
|_|_|_|_|_|_|_|,
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and here is one of the a(7) = 77 ways to tile it according to our rules (the two black squares in the center are identified with X):
_
_____|X|_____
|_|_|___|_|___|. - Greg Dresden, Sep 06 2024
|X|
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