For n ≥ 1, let a_n count the number of ways one can tile a 1 × n chessboard using 1 × 1 square tiles, which come in w colors, and 1 × 2 rectangular tiles, which come in t colors. The results for a_n generalize the Fibonacci numbers and provide generalizations of many of the properties satisfied by the Fibonacci and Lucas numbers. We count the total number of 1 × 1 square tiles and 1 × 2 rectangular tiles that occur among the a_n tilings of the 1 × n chessboard. Further, for these a_n tilings, we also determine: (i) the number of levels, where two consecutive tiles are of the same size; (ii) the number of rises, where a 1 × 1 square tile is followed by a 1 × 2 rectangular tile; and, (iii) the number of descents, where a 1 × 2 rectangular tile is followed by a 1 × 1 square tile. Wrapping the 1 × n chessboard around so that the nth square is followed by the first square, the numbers of 1 × 1 square tiles and 1 × 2 rectangular tiles, as well as the numbers of levels, rises, and descents, are then counted for these circular tilings.