A classic problem in elementary combinatorics asks in how many ways one can tile a 1 × n chessboard using 1 × 1 and 1 × 2 squares. The number of such tilings is the nth Fibonacci number. In a 2010 paper, Grimaldi generalized this problem by allowing for 1 × 1 tiles to have one of w types and 1 × 2 tiles to have one of t types and found that the solutions, in w and t, satisfied many of the same properties as do the Fibonacci and Lucas numbers. In this paper, we present a variant of this generalization. Namely, we require that no two adjacent 1 × 1 tiles be of the same type. This restriction leads to interesting combinatorial connections with coordination sequences and problems in enumerative set theory. We explore these connections, as well as some formulas for generating functions for the numbers of these types of tilings.