T-Tetrominos in Arithmetic Progression

A famous result of D. Walkup is that an $m\times n$ rectangle may be tiled by T-tetrominos if and only if both $m$ and $n$ are multiples of 4. The "if" portion may be proved by tiling a $4\times 4$ block, and then copying that block to fill the rectangle; but, this leads to regular, periodic tilings. In this paper we investigate how much "order" must be present in every tiling of a rectangle by T-tetrominos, where we measure order by length of arithmetic progressions of tiles.