Question

We want to tile a 2×n strip with 2×2 square tiles that are yellow and green, 2×2 L-shaped tiles that are blue and red and 1×1 orange tiles. Give a formula for the number T

n
​
of such tilings. Your solution must consist of a recurrence equation for T
n
​
with a complete justification, followed by a solution of this recurrence. Showing your work.

Answer 1

To find the number of tilings for a 2xn strip, we can use a recurrence equation. The equation is Tn = Tn-1 + Tn-2, with initial conditions T0 = 1 and T1 = 1. This can be solved using the Fibonacci sequence.

Step-by-step explanation:

To find a formula for the number of tilings, we can consider the first tile that is placed in the strip. There are two possible cases:

1. If the first tile is a 2x2 square or an L-shaped tile, then the remaining strip can be tiled in Tn-2 ways (since it reduces the width of the strip by 2).

2. If the first tile is a 1x1 orange tile, then the remaining strip can be tiled in Tn-1 ways (since it reduces the width of the strip by 1).

Therefore, the recurrence equation for Tn is Tn = Tn-1 + Tn-2 for n ≥ 2, with initial conditions T0 = 1 and T1 = 1. Using this recurrence, we can solve for Tn using the Fibonacci sequence.