Advanced handouts

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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Generating Functions/parts/03 coins.tex

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+\section{Coins}
+
+
+Consider the following problem:
+
+\say{How many different ways can you make change for \$0.50 \par
+using pennies, nickels, dimes, quarters and half-dollars?}
+
+\vspace{2mm}
+
+Most ways of solving this involve awkward brute-force
+approache that don't reveal anything interesting about the problem:
+how can we change our answer if we want to make change for
+\$0.51, or \$1.05, or some other quantity?
+
+\vspace{2mm}
+
+We can use generating functions to solve this problem in a general way.
+
+\definition{}
+Let $p_0, p_1, p_2, ...$ be such that $p_k$ is the number
+of ways to make change for $k$ cents with only pennies.
+
+Similarly, let...
+\begin{itemize}
+	\item $n_k$ be the number of ways to make change for $k$ cents with only nickels;
+	\item $d_k$ be the number of ways using only dimes;
+	\item $q_k$ be the number of ways using only quarters;
+	\item and $h_k$ be the number of ways using only half-dollars.
+\end{itemize}
+
+\problem{}<pcoins>
+Let $p(x)$ be the generating function that corresponds to $p_n$. \par
+Express $p(x)$ as a rational function.
+
+\vfill
+
+\problem{}
+Modify \ref{pcoins} to find expressions for $n(x)$, $d(x)$, $q(x)$, and $h(x)$.
+
+\vfill
+\pagebreak
+
+\definition{}
+Now, let $N(x)$ be the generating function for the sequence
+$n_0, n_1, ...$, where $n_k$ is the number of ways to make
+change for $k$ cents using pennies and nickels.
+
+Similarly, let...
+\begin{itemize}
+	\item let $D(x)$ be the generating function for the sequence using pennies, nickels, and dimes;
+	\item let $Q(x)$ use pennies, nickels, dimes, and quarters;
+	\item and let $H(x)$ use all coins.
+\end{itemize}
+
+\problem{}
+Express $N(x)$ as a rational function.
+
+\vfill
+
+\problem{}
+Using the previous problem, write $D(x)$, then $Q(x)$, then $H(x)$
+as rational functions.
+
+
+\vfill
+
+\problem{}
+Using these generating functions, find recurrence relations for
+the sequences $N_k$, $D_k$, $Q_k$, and $H_k$.
+
+\hint{
+	Your recurrence relation for $N_k$ should refer to the
+	previous values of itself and some values of $p_k$.
+	Your recurrence for $D_k$ should refer to itself and $N_k$;
+	the one for $Q_k$ should refer to itself $D_k$;
+	and the one for $H_k$ should refer to itself and $Q_k$.
+}
+
+\vfill
+
+
+\problem{}
+Using these recurrence relations, fill following table
+and solve the original problem.
+
+\begin{center}
+	\begin{tabular}{ c|cccc|cccc|ccc }
+		$n$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\
+		\hline
+		$p_k$ &&&&&&&&&& \\
+		$N_k$ &&&&&&&&&& \\
+		\hline
+		$D_k$ &&&&&&&&&& \\
+		\hline
+		$Q_k$ &&&&&&&&&& \\
+		$H_k$ &&&&&&&&&&
+	\end{tabular}
+\end{center}
+
+\vspace{1cm}
+\pagebreak