Tilings, Compositions, and Generalizations

For n ≥ 1, let an 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 an 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 an tilings of the 1 × n chessboard. Further, for these an 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. 1 Determining an For n ≥ 1, let an 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. Then for n ≥ 2, we have an = wan−1 + tan−2, a0 = 1, a1 = w. (1) This follows by considering how the last square in the 1 × n chessboard is covered. If we have a 1 × 1 square tile in the nth square, then the preceding n − 1 squares can be covered in an−1 ways. The coefficient w accounts for the number of different colors available for the 1 × 1 square in the nth square. Should the last square be covered (along with the (n− 1)st square) by a 1× 2 rectangular tile, then the preceding n− 2 squares of the 1× n