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Math 192r, Problem Set \#6 \\
(due 10/9/01)
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\begin{enumerate}
\item
For each even integer $n \geq 2$,
we can represent each domino tiling of a 3-by-$n$ rectangle
by a code $(a_1,a_2,\dots,a_n)$,
where $a_k$ is the number of vertical dominos
in the $k$th column (always either 0 or 1).
Note that two different tilings can have the same code;
e.g., for $n=2$
there are three tilings but only two codes
(namely $(0,0)$ and $(1,1)$).
Formulate a conjecture for the number of codes
that occur for general $n$.
\item
Let $a_n$ be the number of domino tilings
of a 4-by-$n$ rectangle, with $n \geq 0$
(we put $a_0 = 1$ by convention).
\begin{itemize}
\item[(a)] Prove that the sequence $a_0,a_1,\dots$
satisfies a linear recurrence relation of order 16 or less.
\item[(b)] Prove that the sequence $a_0,a_1,\dots$
satisfies a linear recurrence relation of order 8 or less.
\item[(c)] Prove that the sequence $a_0,a_1,\dots$
satisfies a linear recurrence relation of order 6 or less.
\end{itemize}
\end{enumerate}
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