Suppose there are n0 chairs lined up in a row in a restraurant. Because ofCOVID social distancing recommendations from health officials the proprietorof the restaurant would prefer that customers not sit immediately next to oneanother. We say a seating arrangement of customers is safe if no two individualssit immediately next to one another. For example, forn= 4 the arrangementwould be safe while the arrangementwould not be safe. In additionto safety considerations the owner would also like to make good use of theseats he has available, and therefore he is interested in maximal safe seatingsof customers. We say an seating arrangement is maximal if it is not possible toadd another customer to the arrangement while still maintaining the property ofsafeness (Note: maximal does not necessarily imply that the maximum possiblenumber of people are seated.) Lets(n) denote the number of maximal safeseating arrangements forn0 seats.

a. Derive a recursive formula fors(n) (Hint: consider the cases where theleftmost chair is occuppied/unoccuppied.)

b. Use the recursive formula for s(n) to computes(n) for n{3,4,...,16}

c. Suppose each chair is occuppied with probability12and that whether eachchair is occuppied or not forms a n collection of mutually independent events. For n {0,1,...,16}compute the probability that the randomseating arrangement is a maximal safe seating arrangement.

d. Use the linear recursion fors(n) obtained in Problem 2 to find a closedform formula fors(n) of the forms(n) =a1rn1+a2rn2+a3rn3whereaiandriare fixed values inRfori= 1,2,3 (Hint: find a way of computings(n)by representing the recursion as the product of a particulardiagnolizable33 matrixMraised to an integer power and a particular 31 columnvector. Simplify by usingMk=PDkP1whereDis a diagonal matrixwhose diagonal entries are the eigenvalues ofM.)

e. Letp(n) denote the probability that a random seating arrangement (asdescribed in (c) of Problem 2) is maximal and safe. Use the result from (a)to find the valueR+such thatp(n)en(Note: we saya(n)b(n)if limna(n)/b(n) = 1.