Question 5: Let n be a positive integer and consider a 1 x n board Bn consisting

Question

Question 5: Let n be a positive integer and consider a 1 x n board Bn consisting of n cells, each one having sides of length one. The top part of the figure below shows B You have an unlimited supply of bricks, which are of the following types (see the bottom part of the figure above): ·There are red (R) and blue(B) bricks, both of which are 1 × 1 cells. We refer to these bricks as squares ·There are green (G) bricks, which are 1 × 2 cells. We refer to these as dominoes. A tiling of the board Bn is a placement of bricks on the board such that the bricks exactly cover B, and no two bricks overlap. In a tiling, a color can be used more than once and some colors may not be used at all. The figure below shows an example of a tiling of B.

Explanation / Answer

T1 =2 (R or B)

T2= 5 ( R, R or B, B or B, R or R, B or G)

T3=12 (RRR, BBB, GR, RG, BG, GB, RRB forms 3, and BBR forms 3)

2n+n/2C12(n-2)+n/2C2 2(n-4)+....n/2Ci2(n-i) (n-i)>=0 and n is even

Tn =

2n+(n+1)/2C12(n-2)+(n+1)/2C22(n-4)+...(n+1)Ci2 (n-i)    (n-i)>1 and n is odd

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