Tiling a Strip with Triangles

In this paper, we examine the tilings of a 2 × n “triangular strip” with triangles. These tilings have connections with Fibonacci numbers, Pell numbers, and other known sequences. We will derive several different recurrences, establish some properties of these numbers, and give a refined count for these tilings (i.e., by the number and type of triangles used) and establish several properties of these refined counts. 1 Tiling with the triangular strip A well known combinatorial interpretation of the Fibonacci numbers (i.e., F (0) = 0, F (1) = 1, and F (n) = F (n − 1) + F (n − 2)) is the tiling of a 2 × (n − 1) rectangular strip with vertical and horizontal dominoes. For example F (5) = 5 and there are 5 tilings of the 2× 4 strip: This interpretation can be used to give combinatorial proofs of many interesting properties of the Fibonacci numbers (see [1]). We want to explore a variation of tiling on a board which shares many similar properties. Namely we want to consider tiling a 2×n “triangular strip”. This consists of a parallelogram constructed from 4n equilateral triangles which has two rows of 2n triangles. As an example, the 2× 6 triangular strip is shown below. There are four types of triangles available for us to use for tiling this strip, i.e., which we will denote as 1, 1−, 2, and 2− respectively, the latter two of which we will sometimes call the large triangles. ∗Department of Mathematics, Iowa State University, Ames, IA 50011, USA {jbodeen, butler, shawnkim, xiyuansu, shenzhiw}@iastate.edu