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Compression Theorems for Periodic Tilings and Consequences
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Arthur T. Benjamin

Department of Mathematics

Harvey Mudd College

Claremont, CA 91711 USA

Alex K. Eustis

Department of Mathematics

UC San Diego

La Jolla, CA 92037 USA

Mark A. Shattuck

Department of Mathematics

University of Tennessee

Knoxville, TN 37996 USA

**Abstract:**

We consider a weighted square-and-domino tiling model obtained by
assigning real number weights to the cells and boundaries of an
*n*-board. An important special case apparently arises when these
weights form periodic sequences. When the weights of an
*nm*-tiling form sequences having period *m*, it is shown that
such a tiling may be regarded as a meta-tiling of length *n* whose
weights have period 1 except for the first cell (i.e., are
constant). We term such a contraction of the period in going from
the longer to the shorter tiling as "period compression". It
turns out that period compression allows one to provide
combinatorial interpretations for certain identities involving
continued fractions as well as for several identities involving
Fibonacci and Lucas numbers (and their generalizations).

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(Concerned with sequences
A000032
and
A000045.)

Received January 27 2009;
revised version received August 2 2009.
Published in *Journal of Integer Sequences*, August 30 2009.

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