Tiling circular $n$-boards with squares and (1,1)-fences

We show that the number of ways to tile a circular $n$-board (an annulus divided into $n$ curved unit square cells) with appropriately curved unit squares and curved (1,1)-fences is a Lucas number squared if $n$ is even and an odd-indexed Lucas number otherwise. A curved (1,1)-fence is composed of two curved unit squares separated by a gap in the shape of another curved unit square. This combinatorial interpretation of the Lucas numbers squared and the odd-indexed Lucas numbers allows us to obtain various identities concerning these sequences in a quick and lucid manner.