Question

Problem 4 A row of n parking spaces is to be filled with cars, trucks, buses, and semis. Cars only take up a single parking space, while trucks require two, and both buses and semis need three spaces 4a: How many ways are there to fill up a row consisting of 1 space? 2 spaces? 3 spaces? 4b: Find a linear recurrence relation for the number of ways to fill up n spaces, and solve to find a closed form formula

Answer 1

parking spaces. Let h be the number of ways to fill a row with n 1. hi1(only one car can be placed), h2 2(either one truck, 2 cars) or 5(3 cars or 1 truck and 1 car(2 cases depending on the order), 1 h3 bus or a semi) 2. We will have three cases depending on what vehicle occupies the last slot Case 1: The last vehicle is a car. In this case we have to fill the first n 1 slots which can be done in h- ways we have to fill Case 2: The last vehicle is a truck. In this cas the first n 2 slots which can be done in h-2 ways Case 3: The last vehicle is a bus or semi. In this case we have to fill the first n 3 slots which can be done in h-3 ways. Since last vehicle can be either a semi or bus we have 2h-3 ways in this case. Combining the above cases we get h h-1h-2+2h-3. Now we proceed to solve this reccurence. We try the solutions of form h a for some E C. We get 3 x2-x-2 0. The roots of this polynomial are 2, w,w2(w is the complex cube root of unity). Therefore any solution of the above recurrence is of form h, c12" +cw" +Cgw2n for some c, , es C. Now we put in the values of h, h, h in the above equation and solve for c, C, es(calculations can be simplified by taking ho 1(only one way to fill with 0 spaces- not fill at all!) and using ho instead of ha) giving us three equations C1+C2C31 2ci cwC3w2 = 1 4c1 CzwC3w 2 giving us c2 + c3, c2 (mE N) Adding all the equations we get ci This gives us if n 3m if n 3m1 2"4 2"4 if n 3m2 2"