Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8 |

Department of Mathematics

Humboldt State University

Arcata, CA 95521

USA

Ralph Grimaldi

Department of Mathematics

Rose-Hulman Institute of Technology

Terre Haute, IN 47803

USA

Silvia Heubach

Department of Mathematics

California State University, Los Angeles

Los Angeles, CA 90032

USA

**Abstract:**

We consider tilings of 2 × *n*, 3 × *n*,
and 4 × *n*
boards with 1 × 1 squares and L-shaped tiles covering an area of
three square units, which can be used in four different orientations.
For the 2 × *n* board, the recurrence relation for the number of
tilings is of order three and, unlike most third order recurrence
relations, can be solved exactly. For the 3 × *n* and
4 × *n*
board, we develop an algorithm that recursively creates the basic
blocks (tilings that cannot be split vertically into smaller
rectangular tilings) of size
3 × *k* and 4 × *k* from which
we obtain the generating function for the total number of tilings. We
also count the number of L-shaped tiles and 1 × 1 squares in all
the tilings of the 2 × *n* and 3 × *n* boards
and determine
which type of tile is dominant in the long run.

(Concerned with sequences A028859 and A077917.)

Received April 21 2006;
revised versions received February 28 2007.
Published in *Journal of Integer Sequences* March 19 2007.

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