<filename>v2/018_problem.tex
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\begin{document}
\rating{2}{1}
Let $U_n$ be the set of sequences of positive integers of length $n$ such that
no substring occurs twice.
\begin{figure}[!h]
  \centering
  \begin{align*}
    (1, 1, 2, 2, 1, 3, 1) &\in U_7\\
    (1, 2, 1, 2, 3) &\not\in U_5 \text{ because } (1, 2) \text{ occurs twice.}\\
    (1, 1, 1) &\not\in U_3 \text{ because } (1, 1) \text{ occurs twice.}
  \end{align*}
  \caption{
    An example and two non-examples of sequences with no repeated substrings.
  }
\end{figure}

\begin{question}
  What is the number of sequences in $U_n$ where the sum of terms is minimized?
\end{question}

\begin{related}
  \item What is the minimum least common multiple of a sequence in $U_n$?
    How many such minimal sequences?
  \item What is the minimum product of a sequence in $U_n$?
    How many such minimal sequences?
  \item What if substrings are considered forward and backward?
  \item What if only substrings of length greater $k$ are considered?
  \item What if any term can appear at most $\ell$ times?
\end{related}

\begin{references}
  \item \url{https://oeis.org/A259280}
\end{references}
\end{document}
