Tiling with Cuisenaire Rods

In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times 1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$ rectangular borders. The number of distinct tilings up to rotational symmetry is considered, and closed form expressions are given, in the case of a square border and in the case of a rectangular border. Finally, the number of distinct tilings up to dihedral symmetry is considered, and a closed form expression is given in the case of a square border.