Please answer all parts in detail

e. [12 marks] 1 have a 2x11! counter that i plan to pave with two kinds of tiles 1:1 square tiles and 2:1 Lshaped tiles as shown at the right. i - started to wonder how many different patterns there might be. ne possibility is diagramed below. l4 Note that symmetries are regarded as different patterns. if we reect the above diagram through a 1vertical or horizontal axis we get a different paving. Your job, as detailed below, is the find the number of pavings for my 2:14 counter. [a] {5 marlts] Letxn he the number of pavings ofa 2m counter. Find a recursive equation for the numbers In. Provide the mitial conditions needed to solve the system. [b] [5 marks) Now use the trains method [the rmethodsee For 414]: to solve the recursive system of[a] and find a formula forrn. You must show your work. i am particularly interested in what strategies you might have used to solve the system of equations [Note: There are a pair ofirrational eigenvalues. These can be technically difficult to work with. But they don't have to be. Don't just pile into the calculations hoping that your truck is tough enough to ride over the rough stuff. Step back and think about how you can use the stnicture of the system to get a smooth and even elegant ride. There are a number of ways to simplify the calculations.] (c) [2 marks] Eheclt your formulae from {b} by putting your recursive equations on a spread sheet and use it to find 1:\". Provide the results of the spreadsheet for n = 1 to 14. Then use the formula to calculate I\" and compare