Jack has an unlimited number of ornamental tiles each comprised of three unit squares. Some tiles are straight 31 tiles, which come in 2 different colours. Maple plot The other tiles are L-shaped tiles, which also come in 2 different colours. Maple plot Let an be the number of ways to completely tile a 2(3n) grid, and let bn be the number of ways to completely tile a 2(3n1) grid whose top-left corner square has been removed. The corresponding grids for a1 and b2 are shown below. Maple plot Notice that tiles may be rotated when placed in the grid, and that two

Jack has an unlimited number of ornamental tiles each comprised of three unit squares. Some tiles are straight 31 tiles, which come in 2 different colours. The other tiles are L-shaped tiles, which also come in 2 different colours. Let an be the number of ways to completely tile a 2(3n) grid, and let bn be the number of ways to completely tile a 2(3n1) grid whose top-left corner square has been removed. The corresponding grids for a1 and b2 are shown below. Notice that tiles may be rotated when placed in the grid, and that two distinct grid tilings are considered different even if they are rotations or reflections of one another. For example, two different tilings of a 23 grid are shown below. (a) Complete the following list of initial values: (Recall that the straight tiles come in 2 different colours, and the L-shaped tiles come in 2 different colours.) a0=a1=b0=0b1= (b) Help Jack find recurrence relations that relate the two sequences. We have an=pbn+qan1 for all n2, where p= and q= and bn=ran1+sbn1 for all n2, where r= and s= (c) Hence find a recurrence relation that relates terms only in the sequence (an). We have an=tan1+uan2 for all n2, where t= andu= (d) The closed formula for an is given by an=A(k1)n+B(k2)nforallnN, where A,B,k1, and k2 are some constants. What are the values of k1 and k2 ? k1,k2=