Tilings, compositions, and generalizations

Grimaldi, Ralph P.

Journal of integer sequences, Tome 13 (2010) no. 6.

Résumé

Summary: For $n \ge 1$, let $a_{n}$ count the number of ways one can tile a $1 \times n$ chessboard using $1 \times 1$ square tiles, which come in $w$ colors, and $1 \times 2$ 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 $1 \times 1$ square tiles and $1 \times 2$ rectangular tiles that occur among the $a_{n}$ tilings of the $1 \times n$ 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 $1 \times 1$ square tile is followed by a $1 \times 2$ rectangular tile; and, (iii) the number of descents, where a $1 \times 2$ rectangular tile is followed by a $1 \times 1$ square tile. Wrapping the $1 \times n$ chessboard around so that the $n$th square is followed by the first square, the numbers of $1 \times 1$ square tiles and $1 \times 2$ rectangular tiles, as well as the numbers of levels, rises, and descents, are then counted for these circular tilings.

Détail

Classification : 11B37, 11B39
Keywords: tilings, Fibonacci numbers, Lucas numbers, jacobsthal numbers, levels, rises, descents, circular tilings