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Social Distancing, Primes, and Perrin Numbers

restaurant owner who is famous for pondering the mathematics behind flipping pancakes (Problem E2569, Amer. Math. Monthly, 82 [1975], 1010). With the COVID-19 pandemic, Harry has become less harried, and he has been thinking about the possible seating arrangements at his restaurant’s large round outdoor table. He loves displaying his spectacular pancake flipping abilities and hopes to have as many people at the table as possible. The table would ordinarily seat 13, but social distancing guidelines state that no two people may sit in adjacent seats. So Harry got to wondering: in how many different ways can the table be full, in the sense that no additional customer is able to sit down without violating the distancing guidelines? For example, both tables shown in figure 1 are full, even though they have different numbers of customers.