  In this paper a closed form expression for the number of tilings of an
$n\times n$ square border with $1\times 1$ and $2\times1$ 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.
