Let a(n) be the number of ways to lay out napkins for n people. Now, suppose you add one more person. You have four colors of napkin to give that person but you cannot give him the napkins of the people on either side. Since those are of different colors (the people were sitting next to each other) you can choose either of the two remaining colors to give him. That is, a(n+ 1)= a(n)+ 2.