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Tilings, Compositions, and Generalizations
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Ralph P. Grimaldi

Department of Mathematics

Rose-Hulman Institute of Technology

Terre Haute, Indiana 47803

USA

**Abstract:**

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 *n*th 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.

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(Concerned with sequences
A000032
A000045
A000129
A001045.)

Received August 13 2009;
revised version received June 15 2010.
Published in *Journal of Integer Sequences*, June 16 2010.

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