Compression Theorems for Periodic Tilings and Consequences

被引：0

作者：

Benjamin, Arthur T. ^{[1
]}

Eustis, Alex K. ^{[2
]}

Shattuck, Mark A. ^{[3
]}

机构：

[1] Harvey Mudd Coll, Dept Math, Claremont, CA 91711 USA

[2] Univ Calif San Diego, Dept Math, La Jolla, CA 92037 USA

[3] Univ Tennessee, Dept Math, Knoxville, TN 37996 USA

来源：

JOURNAL OF INTEGER SEQUENCES | 2009年 / 12卷 / 06期

关键词：

continued fraction; polynomial generalization; Fibonacci number; Lucas number; tiling;

D O I：

暂无

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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).

引用

页数：15