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\begin{center}
\vskip 1cm{\LARGE\bf 
Polygonal Chain Sequences \\
\vskip .1in
in the Space of Compact Sets
}
\vskip 1cm
\large
Steven Schlicker \\
Grand Valley State University \\
Allendale, MI 49401 \\
USA \\
\href{mailto:schlicks@gvsu.edu}{\tt schlicks@gvsu.edu}\\
\ \\
Lisa Morales \\
University of California, Riverside \\
Riverside, CA 92521 \\
USA \\
\href{mailto:lmora008@student.ucr.edu}{\tt lmora008@student.ucr.edu} \\
\ \\
Daniel Schultheis \\
University of California, San Diego \\
La Jolla, CA 92093 \\
USA \\
\href{mailto:dschulth@math.ucsd.edu}{\tt dschulth@math.ucsd.edu} \\
\end{center}

\vskip .2in

\begin{abstract}
Configurations in the hyperspace of all non-empty compact subsets of
$n$-dimensional real space provide a potential wealth of examples of
familiar and new integer sequences. For example, Fibonacci-type
sequences arise naturally in this geometry. In this paper, we introduce
integer sequences that are determined by polygonal chain
configurations.
\end{abstract}

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\section{Introduction} 

The Hausdorff metric $h$ provides a way of measuring the distance between nonempty closed and bounded subsets of $\mathbb{R}^n$ using the standard Euclidean metric $d_E$. The metric $h$ imposes a geometry on $\mathcal{H}(\mathbb{R}^n)$, the hyperspace of all non-empty, compact subsets of $\mathbb{R}^n$. The concept of betweenness in this geometry gives rise to interesting examples of integer sequences. In this context we will introduce polygonal chains and new integer sequences related to them. Throughout this article, we let $F_{k}$ denote the $k^{\text{th}}$ Fibonacci number (with $F_0 = 0, F_1=1$) and $L_k$ the $k^{\text{th}}$ Lucas number (with $L_0=2, L_1=1$). Recall that the Lucas numbers $L_k$ satisfy 
\[L_k = F_{k-1} + F_{k+1} = 2F_{k-1} + F_k.\]

\section{The Hausdorff Metric}
The Hausdorff metric $h$ was introduced by Felix Hausdorff in the early 20th century as a way to measure the distance between compact sets. The Hausdorff metric imposes a geometry on the space $\mathcal{H}(\mathbb{R}^n)$, which will be the subject of our study. To distinguish between $\mathbb{R}^n$ and $\mathcal{H}(\mathbb{R}^n)$, we will refer to \emph{points} in $\mathbb{R}^n$ and \emph{elements} in $\mathcal{H}(\mathbb{R}^n)$.

\begin{definition} \label{def:Hausdorffmetric}
Let $A$ and $B$ be elements in $\mathcal{H}(\mathbb{R}^n)$. The \emph{Hausdorff distance}, $h(A,B)$, between $A$ and $B$ is 
\[h(A,B) = \max \{d(A,B), d(B,A)\},\]
where 
$d(A,B) = \max_{a \in A} \{d(a,B) \}$ and $d(a,B) = \min_{b \in B} \{d_E(a,b) \}$.
\end{definition}

As an example, let $A$ be the two point set $\{(-1,0), (1,0)\}$ and $B$ the circle centered at (0,0) of radius 1 in $\mathbb{R}^2$. Since $A \subset B$, we have $d(A,B)=0$. However, $d(B,A) = d_E((0,1),(1,0)) = \sqrt{2}$. Note that $d(A,B) \neq d(B,A)$, so the function $d$ is not symmetric. That is why we need the maximum of $d(A,B)$ and $d(B,A)$ to obtain a metric. In our example, we have $h(A,B) = d(B,A) = \sqrt{2}$. See Barnsley \cite{Barnsley} for a proof that $h$ is a metric on $\mathcal{H}(\mathbb{R}^n)$. The corresponding metric space, $(\mathcal{H}(\mathbb{R}^n), h)$, is then itself a complete metric space \cite{Barnsley}.  

\section{Betweenness in $\mathcal{H}(\mathbb{R}^n)$}
Betweenness will play a major role in this paper, allowing us to find new integer sequences. In this section we define ``betweenness" in $\mathcal{H}(\mathbb{R}^n)$, mimicking the idea of betweenness in $\mathbb{R}^n$ under the Euclidean metric. It is in this context that we will later define polygonal chains and introduce an infinite family of new integer sequences. Before we discuss betweenness, we define the dilation of a set, which will play a critical role in what follows. 

\begin{definition} Let $A \in \mathcal{H}(\mathbb{R}^n)$ and let $s&gt;0$ be a real number. The \emph{dilation} of $A$ by $s$ is the set 
\[(A)_s = \{x \in \mathbb{R}^n : d_E(x,a) \leq s \text{ for some } a \in A \}.\]
\end{definition}
As a simple example, the dilation of a single point set $A = \{a\}$ by $s$ is the closed ball centered at $a$ of radius $s$. In general, the $s$-dilation of the set $A$ is the union of all closed $s$ balls centered at points in $A$. Using dilations, we can alternatively define $h(A,B)$ as the minimum value of $s$ so that the $s$-dilation of $A$ encloses $B$ and the $s$-dilation of $B$ encloses $A$. An important and useful result about dilations is the following (Theorem 4 from the article \cite{BMMS}).

\begin{theorem} \label{thm:dilations}
Let $A \in \mathcal{H}(\mathbb{R}^n)$ and let $s&gt;0$.  Then $(A)_s$ is a compact set with $h(A, (A)_s)=s$. Moreover, if $ C \in \mathcal{H}(\mathbb{R}^n)$ and $h(A,C) \leq s$, then $C \subseteq (A)_s$.  
\end{theorem}

In other words, the element $(A)_s$ is the largest element (in the sense of containment) that is a distance $s$ from $A$. Now we discuss betweenness. In the standard Euclidean geometry, a point $x$ lies between the points $a$ and $b$ if and only if  $d_E(a,b) = d_E(a,x) + d_E(x,b)$. We extend this idea to $\mathcal{H}(\mathbb{R}^n)$.

\begin{definition} \label{def4}
Let $A, B \in \mathcal{H}(\mathbb{R}^n)$ with $A \neq B$. The element $C \in \mathcal{H}(\mathbb{R}^n)$ \emph{lies between} $A$ and $B$ if $h(A,B) = h(A,C) + h(C,B)$.
\end{definition}

\begin{figure}[t]
\begin{center}
\resizebox{!}{1.5in}{\includegraphics{Between}}
\caption{\scriptsize Infinitely many elements at the same location between $A$ and $B$.}
\label{fig:Between}
\end{center}
\end{figure}
Notation: We write $ACB$ to signify that $C$ is between $A$ and $B$ as in Blumenthal \cite{Blumenthal}. Betweenness in $\mathbb{R}^n$ is quite different than betweenness in $\mathcal{H}(\mathbb{R}^n)$. Consider the following examples.
\begin{example} \label{ex:inf_between} Let $A$ the union of the origin with the circle centered at (0,0) with radius 10 and $B$ the circle centered at (0,0) with radius 5 in $\mathcal{H}(\mathbb{R}^2)$ as shown in Figure \ref{fig:Between}. Then $h(A,B) = d(A,B) = d(B,A) = 5$. The union $C \cup C^*$ of the two dashed circles is the element $(A)_2 \cap (B)_3$. Theorem \ref{thm:dilations} shows that any element $X$ satisfying $AXB$ with $h(A,X) = s$ and $h(B,X) = h(A,B)-s$ must be a subset of $(A)_s \cap (B)_{h(A,B)-s}$. It is not difficult to see that if $X_c = \{c\} \cup C^*$ where $c$ is any point on the circle $C$, then $X$ satisfies $AXB$ with $h(A,X) = 2$. So in this example, there are infinitely many different elements $X$ satisfying $AXB$ and $h(A,X)=2$. 
\end{example} 
\begin{example} \label{ex:finite_between} Let $A = \{(-2,0), (0,0), (2,0)\}$ and $B = \{(-1,0), (1,0)\}$ in $\mathbb{R}^2$ as shown in Figure \ref{fig:Between2}. Let $0 &lt; s &lt; h(A,B)$ and let $t = h(A,B)-s$. In this case, the set $C = (A)_s \cap (B)_{t}$ is the four point set $C = \{c_1, c_2, c_3, c_4\}$. Since any element $C'$ that lies between $A$ and $B$ at the distance $s$ from $A$ is a subset of $C$, there are at most 16 elements $C'$ that could satisfy $AC'B$ with $h(A,C') = s$. However, not every subset of $C$ has this property. For example, if $C' = \{c_1\}$, then $h(A,C') = 3 + t$ and $C'$ does not satisfy $AC'B$. We leave it to the reader to show that the sets $C$, $\{c_1, c_2, c_4\}$ and $\{c_1, c_3, c_4\}$ are the only sets that lie between $A$ and $B$ at the distance $s$ from $A$. It is this situation in which we have only finitely many elements between two fixed elements at a specific distance from one of them that will be of interest to us. 
\end{example}
The next definition formalizes what is meant when we refer to two elements at the \textit{same location} between sets $A$ and $B$.

\begin{definition} \label{def:location}
Let $A, B \in \mathcal{H}(\mathbb{R}^n)$ with $A \neq B$ and let $C,C' \in \mathcal{H}(\mathbb{R}^n)$ satisfy $ACB$ and $AC'B$.  The elements $C,C'$ are said to be \emph{at the same location} between $A$ and $B$ if $h(A,C)=h(A,C')=s$ for some  $0&lt; s &lt; h(A,B)$.
\end{definition}   

\begin{figure}[t]
\begin{center}
\resizebox{!}{1.0in}{\includegraphics{Between2}}
\caption{\scriptsize Exactly 3 different elements at the same location between $A$ and $B$.}
\label{fig:Between2}
\end{center}
\end{figure}

\section{Finding Points Between $A$ and $B$} \label{sec:Between}

Let $A, B \in \mathcal{H}(\mathbb{R}^n)$. If a set $C \in \mathcal{H}(\mathbb{R}^n)$ satisfies $ACB$ with $h(A,C)=s$, then it follows from Theorem \ref{thm:dilations} that $C \subseteq (A)_s \cap (B)_{h(A,B)-s}$. As we saw in Examples \ref{ex:inf_between} and \ref{ex:finite_between}, for some elements $A$ and $B$ there are infinitely many different elements at the same location between $A$ and $B$ and for other $A$ and $B$ there are only finitely many elements at the same location between them. The latter situation is of interest to us. In article \cite{BLSSZ}, the authors show that in order for a pair of elements $A$ and $B$ in $\mathcal{H}(\mathbb{R}^n)$ to have a finite number of elements in $\mathcal{H}(\mathbb{R}^n)$ at a given location between $A$ and $B$, every point in $A$ must be a common distance $r=h(A,B)$ away from $B$ (that is, $d(a,B) = r$ for all $a \in A$) and every point in $B$ must be that same distance $r$ from $A$ ($d(b,A)=r$ for all $b \in B$). If this condition is not met, there will be infinitely many elements in $\mathcal{H}(\mathbb{R}^n)$ at each location between $A$ and $B$. If a pair of elements $A,B \in \mathcal{H}(\mathbb{R}^n)$ satisfies this condition, we call the pair a configuration. 

\begin{definition} A \emph{finite configuration} is a pair $[A,B]$ of finite sets $A, B \in \mathcal{H}(\mathbb{R}^n)$ where $d(a,B) = d(b,A) = h(A,B)$ for all $a \in A$ and $b \in B$.
\end{definition}

In article \cite{BLSSZ}, the authors also show that if $[A,B]$ is a finite configuration, then the number of elements $C$ satisfying $ACB$ and $h(A,C)=s$ is independent of $s$ for all $0 &lt; s &lt; h(A,B)$. Thus, we denote by $\#([A,B])$ the number of elements between $A$ and $B$ in a finite configuration $[A,B]$ at any distance $s$.  

Note that if $[A,B]$ is a finite configuration, then $C = (A)_s \cap (B)_{h(A,B)-s}$ will be a finite set for any $0 &lt; s &lt; h(A,B)$, so there are only finitely many subsets of $C$ and $\#([A,B])$ must be finite by Theorem \ref{thm:dilations} (see Figure \ref{fig:Between2} for an example). There are infinite configurations, but in this paper we will only be concerned with finite ones. 

\section{String and Polygonal Configurations}

Two basic configurations that will be relevant to our work are string and polygonal configurations. A string configuration is any finite set of points equally spaced on a line segment, with sets $A$ and $B$ consisting of alternate points on the line \cite{LSS}. Figure \ref{fig:Between2} is an example. If $|A \cup B| = k$, then we will call the string configuration $[A,B]$ a $k-string$ and denote it as $S_k$. So Figure \ref{fig:Between2} shows a 5-string or $S_5$. In article \cite{LSS} (Theorem 6.1) the authors show  
\[\#(S_k) = F_{k-1}.\]
In Figure \ref{fig:Between2}, the elements $\{c_1, c_2, c_3, c_4\}, \{c_1, c_2, c_4\}$, and $\{c_1, c_3, c_4\}$ are the elements that satisfy $ACB$ at the indicated location in this example. This is as expected, since $\#(S_5) = F_4 = 3$.   

Another basic family of configurations is the collection of polygonal configurations. Let $A$ and $B$ be the alternate vertices of a regular $2m$-gon with $m \in \mathbb{N}$. We denote the configuration $[A,B]$ in this case by $P_m$ and call this a polygonal configuration \cite{LSS}. An example of $P_4$ is shown in Figure \ref{fig:P_4}, with the set $A$ as the filled points and the set $B$ as the open points. Line segments are drawn between points to indicate the pairs of points $a \in A$ and $b \in B$ satisfying $d_E(a,b) = h(A,B)$. These points will be important in what follows.

\begin{figure}[t]
\begin{center}
\resizebox{!}{1.0in}{\includegraphics{P_4}}
\caption{\scriptsize The $P_4$ configuration.}
\label{fig:P_4}
\end{center}
\end{figure}

\begin{definition} Let $[A,B]$ be a finite configuration. The points $a \in A$ and $b \in B$ are \emph{adjacent} if $d_E(a,b) = h(A,B)$. If $a \in A$, the \emph{adjacency set} of $a$, $[a]_B$, is defined as $[a]_B = \{b \in B : d_E(a,b) = h(A,B)\}$. 
\end{definition}

Again in the paper \cite{LSS} (Theorem 6.3) the authors show 
\[\#(P_m) = F_{2m} + 2 F_{2m-1} = L_{2m}.\]

String and polygonal configurations form the building components of polygonal chains.

\section{Polygonal Chains}

A polygonal chain $PC_{m,l}^{k}$ is a configuration obtained by connecting $k$ copies of $P_m$ at antipodal points with string configurations of a fixed length $l+1$. When $k=1$, we define $PC_{m,l}^1$ to be $P_m$. Examples of polygonal chains are shown in Figure \ref{fig:Chain_examples}. In this section we determine recursive formulas to find $\#(PC_{m,l}^{k})$. 

\begin{figure}[t]
\begin{center}
\resizebox{!}{0.4in}{\includegraphics{P2_3_S3}} \\

\vspace{10pt}

\resizebox{!}{0.65in}{\includegraphics{P3_2_S2}}
\caption{\scriptsize Top: $PC_{2,3}^3$ \hspace{0.25in} Bottom: $PC_{3,2}^2$.}
\label{fig:Chain_examples}
\end{center}
\end{figure}

\subsection{Adjoining Configurations} \label{subsec:adjoin}

To compute $\#(PC_{m,l}^{k})$, we will ultimately reduce the problem to one of determining $\#(X)$ for smaller configurations $X$ created from adjoins of polygonal and string configurations. Before discussing adjoins, we need to understand equivalent configurations. The idea here is that in the computation of $\#([A,B])$ the distance between adjacent points is irrelevant - only adjacencies matter. 

\begin{definition} The finite configuration $[X,Y]$ is \emph{equivalent} to the finite configuration $[A,B]$ if there are bijections $f : A \to X$ and $g: B \to Y$ such that for all $a \in A$ and $b \in B$, we have $a$ adjacent to $b$ if and only if $f(a)$ is adjacent to $g(b)$. When $[X,Y]$ is equivalent to $[A,B]$ we write $[X,Y] \sim [A,B]$.
\end{definition} 

It is easy to show that the relation $\sim$ is an equivalence relation on the set of finite configurations. One important result involving equivalent configurations is that if $[X,Y]$ and $[A,B]$ are equivalent configurations, then $\#([X,Y]) = \#([A,B])$ \cite{BLSSZ} (Theorem 4.1). Now we adjoin a point to a configuration. 

\begin{definition} Let $[A,B]$ be a finite configuration. The configuration $[A,B](a,y)$ obtained by \emph{adjoining a point} $y \not\in A \cup B$ to $[A,B]$ at the point $a \in A$ is the configuration $[A,B']$, where $B' = B \cup \{y\}$, $d_E(y,a)=h(A,B)$, and $d_E(y,x) &gt; h(A,B)$ for all $x \in A \cup B$ with $x \neq a$. 
\end{definition}

For example, Figure \ref{fig:P_4_adjoin} shows the configuration obtained by adjoining a point $y$ to $P_4$ at point $a$. 

\begin{figure}[b]
\begin{center}
\resizebox{!}{0.85in}{\includegraphics{P_4_adjoin}}
\caption{\scriptsize Adjoining a point to $P_4$.}
\label{fig:P_4_adjoin}
\end{center}
\end{figure}

One tool for computing $\#([A,B](a,y))$ for a finite configuration $[A,B]$ is the following \cite{LSS} (Theorem 6.2).

\begin{theorem} \label{thm:adjoin} Let $[A,B]$ be a finite configuration. Define a new configuration $X'$ by adjoining a point $y$ to $[A,B]$ at the point $a \in A$. Furthermore, assume $a$ is adjacent to $k$ points in $B$, each of which is adjacent to at least one more point in $A$. Then 
\[\#(X') = \#([A,B]) + \#([A - \{a\}, B]).\] 
\end{theorem}

As an example, if $[A,B] = P_4$ and $X' = [A,B](a,y)$ as shown in Figure \ref{fig:P_4_adjoin}, then $\#(X') = \#(P_4) + \#(S_7) = L_8 + F_6 = 55$. 

\begin{figure}[t]
\begin{center}
\resizebox{!}{0.75in}{\includegraphics{P_2_adjoin_P_3}}
\caption{\scriptsize Adjoining $P_2$ and $P_3$.}
\label{fig:P_2_adjoin_P_3}
\end{center}
\end{figure}

We can also adjoin two configurations. 

\begin{definition} Let $[A,B]$ and $[X,Y]$ be finite configurations and let $a \in A$ and $y \in Y$. Let $[X',Y']$ be a configuration equivalent to $[X,Y]$ through  bijections $f:X \to X'$ and $g:Y \to Y'$ so that 
\begin{enumerate}
\item $h(X',Y') = h(A,B)$, 
\item $d_E(g(y),a) = h(A,B)$, 
\item $d(z,A \cup B) &gt; h(A,B)$ for all $z \in X' \cup Y'$, $z \neq g(y)$, and
\item $d(w, X' \cup Y') &gt; h(A,B)$ for all $w \in A \cup B$, $w \neq a$.
\end{enumerate}
The \emph{adjoin} $[A,B][a] \oplus [X,Y][y]$ of the configurations at the points $a$ and $y$ is the configuration $[A \cup X', B \cup Y']$. 
\end{definition}

As an example, the adjoin of $P_2$ with $P_3$ at the points $a$ and $y$ is shown in Figure \ref{fig:P_2_adjoin_P_3}.

\begin{figure}[b]
\begin{center}
\resizebox{!}{0.75in}{\includegraphics{S1_P3_S1}}
\caption{\scriptsize $P_3 \oplus \{S_1[1]; S_1[4]\}$.}
\label{fig:S1_Pm_S1}
\end{center}
\end{figure}

Notation: In this paper, we will consider configurations obtained by adjoining string configurations at their endpoints to polygonal configurations. If we adjoin a single string configuration to a polygonal configuration, the point of adjoin does not matter due to symmetry. The notation we use for adjoining a string configuration $S_l$ to a polygonal configuration $P_m$ is $P_m \oplus S_l$. There are cases in which we will adjoin more than one string configuration to a polygonal configuration. In these situations, we will label the vertices of a polygonal configuration counterclockwise in order (the starting point does not matter due to symmetry). The notation we will use to adjoin string configurations of length $s$ and $t$ to a polygonal configuration $P_m$ at the points labeled $i$ and $j$, respectively is $P_m \oplus \{S_s[i], S_t[j]\}$. An illustration of $P_3 \oplus \{S_1[1]; S_1[4]\}$ is shown in Figure \ref{fig:S1_Pm_S1}. 

The next theorem shows us a method for computing $\#(X)$ by breaking $X$ into smaller disconnected configurations. A configuration $[A,B]$ is connected if, given $a \in A$ and $b \in B$, there is a sequence $\{x_k\}_{k=1}^m \in A \cup B$ such that $x_1 = a$, $x_m = b$ and $x_i$ is adjacent to $x_{i+1}$ for each $i$ \cite{BLSSZ}. In other words, there is a string in $A \cup B$ connecting $a$ to $b$. A configuration is disconnected if it is not connected. The proof of Theorem \ref{thm:loop_algorithm} depends on the next result from the paper \cite{BLSSZ} (Theorem 6.1).

\begin{theorem} \label{thm:basic_facts} Let $[A,B]$ be a configuration and let $r = h(A,B)$. If $[X,Y]$ is a configuration with $h(A,B) = h(X,Y)$, $d_E(a,y) &gt; h(A,B)$ for all $a \in A$ and $y \in Y$, and $d_E(b,x) &gt; h(A,B)$ for all $b \in B$ and $x \in X$, then $\#([A \cup X, B \cup Y]) = \#([A,B]) \#([X,Y])$.
\end{theorem}

In other words, if $[U,V]$ is a disconnected configuration constructed from configurations $[A,B]$ and $[X,Y]$, with $U = A \cup X$ and $V = B \cup Y$, then $\#([U,V]) = \#([A,B]) \#([X,Y])$. The next theorem describes another important computation technique. 

\begin{theorem} \label{thm:loop_algorithm} 
Let $[A,B]$ and $[X, Y]$ be finite configurations and let $[U,V] = [A,B][a] \oplus [X,Y][y]$ for some $a \in A$ and $y \in Y$. Then
\[\#([U,V]) = \#([A,B]) \ \#([X,Y]) + \#([A \oplus S_{1}[a], B]) \ \#[X, Y \oplus S_{1}[y]].\]
\end{theorem}

\begin{proof} Without loss of generality, we assume $h(A,B) = h(X,Y)$. Let $s,t &gt; 0$ with $s+t = h(A,B)$. Let $C = (A)_s \cap (B)_t$ be the largest element that lies between $A$ and $B$ at a distance $s$ from $A$, and let $Z = (X)_s \cap (Y)_t$ be the largest element that lies between $X$ and $Y$ at a distance $s$ from $X$. Then in the configuration $[U,V]$, the largest element between $A \cup X$ and $B \cup Y$ will be $C \cup Z \cup \{c_{0}\}$, where $c_{0}$ is the point which lies between the newly adjacent points $a$ and $y$. See Figure \ref{fig:P_2_adjoin_P_3} for an illustration.

We now consider which subsets $D$ of $C \cup Z \cup \{c_{0}\}$ satisfy $UDV$. Since $D$ satisfies $UDV$, we know that $[a']_{D} \neq \emptyset$, $[b']_{D} \neq \emptyset$, $[x']_{D} \neq \emptyset$, and $[y']_{D} \neq \emptyset$ for all $a' \in A, b' \in B, x' \in X$, and $y' \in Y$. We consider two cases.

\begin{description}
\item[Case 1] $c_{0} \not \in D$. In this case, let $C' = \{d \in D : d \in [a']_D \text{ for some } a' \in A\}$ and let $Z' = D - C'$. By construction, if $c' \in C'$, then $c' \neq c_0$ and so $c' \in C$. Thus $C' \subseteq C$. If $z' \in Z'$, then $z' \neq c_0$ and $z' \in Z$. Now, if $a' \in A$ and $b' \in B$, we know $[a']_D \neq \emptyset$ and $[b']_D \neq \emptyset$. Therefore, $[a']_{C'} \neq \emptyset$ and $[b']_{C'} \neq \emptyset$. So $C'$ satisfies $AC'B$. Similarly, $Z'$ satisfies $XZ'Y$. By Theorem \ref{thm:basic_facts}, the number of such sets $D = C' \cup Z'$ is $\#([A,B]) \ \#([X,Y])$.
\item[Case 2] $c_{0}\in D$. In this case, we have $c_{0} \in [a]_{D}$ and $c_{0} \in [y]_{D}$. The existence of the point $c_0$ in $D$ allows for the potential removal of other points in $D$ that are adjacent to $a$ or $y$. In other words, $D = C' \cup \{c_0\} \cup Z'$, where $C'$ satisfies $AC'(B \oplus S_1[a])$ and $Z'$ satisfies $(X \oplus S_1[y])Z'Y$.  The number of such sets $D$ is $\#([A \oplus S_1[a],B]) \cdot \#(X, Y \oplus S_{1}[y]])$.
\end{description}
By combining the results of the two cases here, we arrive at the expected formula $\#([U,V]) = \#([A,B]) \ \#([X,Y]) + \#([A \oplus S_{1}[a], B]) \ \#[X, Y \oplus S_{1}[y]]$.
\end{proof}

As an example, consider the configuration $X$ obtained by adjoining $P_3$ and $P_2$ as shown in Figure \ref{fig:P_2_adjoin_P_3}. Theorem \ref{thm:loop_algorithm} shows 
\[\#(X) = \#(P_3) \ \#(P_2) + \#(P_3 \oplus S_1) \ \#(P_2 \oplus S_1).\]
Now Theorem \ref{thm:adjoin} gives us $\#(P_3 \oplus S_1) = \#(P_3) + \#(S_5) = 18 + 3 = 21$ and $\#(P_2 \oplus S_1) = \#(P_2) + \#(S_3) = 7 + 1 = 8$. So 
\[\#(X) = (18)(7) + (21)(8) = 294.\]

\section{Finding $\#(PC_{m,l}^{k})$}

In this section, we determine a pair of recursive formulas to find $\#(PC_{m,l}^{k})$. As the first step in our computations, we determine $\#(X \oplus S_t)$ for $t \geq 2$ for any finite configuration $X$ in terms of $\#(X)$ and $\#(X \oplus S_1)$. 

\begin{theorem} \label{thm:config_plus_string2} Let $X$ be a finite configuration and $t \geq 2$. Then 
\[\#(X \oplus S_t) = F_t \#(X \oplus S_1) + F_{t-1} \#(X).\]
\end{theorem}

\begin{proof} Using Theorem \ref{thm:adjoin}, we see that our theorem is certainly true with $t=2$ and $t=3$. We proceed by induction and suppose our theorem is true for all $s$, $1 \leq s \leq t$. Then 
\begin{align*}
\#(X \oplus S_{t+1}) &amp;= \#(X \oplus S_t) + \#(X \oplus S_{t-1})  \\
	&amp;= \left[ F_t \#(X \oplus S_1) + F_{t-1} \#(X) \right] + \left[ F_{t-1} \#(X \oplus S_1) + F_{t-2} \#(X) \right] \\
	&amp;= (F_t + F_{t-1}) \#(X \oplus S_1) + (F_{t-1} + F_{t-2}) \#(X) \\
	&amp;=  F_{t+1} \#(X \oplus S_1) + F_{t} \#(X).
\end{align*}
\end{proof}

\begin{figure}[t]
\begin{center}
\resizebox{!}{0.5in}{\includegraphics{Plus_S1}}
\caption{\scriptsize $PC_{2,3}^3 \oplus S_1$.}
\label{fig:Plus_S1}
\end{center}
\end{figure}

By $PC_{m,l}^{k} \oplus S_t$ we mean the adjoin of $S_t$ with $PC_{m,l}^{k}$ at a point antipodal to the connected strings. An example of $PC_{2,3}^3 \oplus S_1$ is shown in Figure \ref{fig:Plus_S1}. A direct consequence of Theorem \ref{thm:config_plus_string2} is 

\begin{corollary} \label{cor:Pm_Sl_plus_St2} For $m \geq 2$, $k \geq 1$, $l \geq 1$, and $t \geq 2$, 
\[\#(PC_{m,l}^{k} \oplus S_t) = F_t \#(PC_{m,l}^{k} \oplus S_1) + F_{t-1} \#(PC_{m,l}^{k})\]
\end{corollary}

Now we add in the case when $t=1$ for polygonal configurations. Note first that  
\[L_{2m} + F_{2m-2} = (F_{2m-2} + F_{2m-1}) + F_{2m+1} = F_{2m} + F_{2m+1} = F_{2m+2}.\]

\begin{corollary} \label{cor:Pm_plus_St} For $m \geq 2$ and $t \geq 1$, 
\begin{equation*} \label{eq:Pm_plusS1}
\#(P_m \oplus S_t) = F_t F_{2m+2} + F_{t-1} L_{2m}. 
\end{equation*}
\end{corollary}

\begin{proof} When $t=1$, a direct application of Theorem \ref{thm:adjoin} gives us
\[\#(P_m \oplus S_1) = \#(P_m) + \#(S_{2m-1}) = L_{2m} + F_{2m-2} = F_{2m+2} = F_1 F_{2m+2} + F_0 L_{2m}.\]
For $t \geq 2$, we use Theorem \ref{thm:config_plus_string2} to obtain
\[\#(P_m \oplus S_t) = F_t \#(P_m \oplus S_1) + F_{t-1} \#(P_m) = F_t F_{2m+2} + F_{t-1} L_{2m}.\]
\end{proof}

We need one more preliminary calculation before we determine $\#(PC_{m,l}^{k})$.

\begin{lemma} \label{lem:S1_Pm_S1} For $m \geq 2$, 
\[\#(P_m \oplus \{S_1[1]; S_1[m+1]\}) = F_{m+2}^2.\]
\end{lemma}

\begin{proof} Figure \ref{fig:S1_Pm_S1} shows an example of the configurations in question. By Theorem \ref{thm:adjoin} and Corollary \ref{cor:Pm_plus_St},  
\begin{align*}
\#(P_m \oplus \{S_1[1]; S_1[m+1]\}) &amp;= \#(P_m \oplus S_1) + \#(S_{2m-1} \oplus S_1[m]) \\
	&amp;= F_{2m+2} + \#(S_{2m-1}) + \#(S_{m-1})^2 = F_{2m+2} + F_{2m-2} + F_{m-2}^2.
\end{align*}
We use the identity 
\[F_{2m} = F_{m+1}^2 - F_{m-1}^2\]
for $m \geq 1$ (see \cite{Koshy}, Corollary 5.4 or \cite{BQ}, Identity 14 (note that $f_m = F_{m+1}$)) to obtain  
\begin{align*}
F_{2m+2} + F_{2m-2} + F_{m-2}^2 &amp;= F_{2(m+1)} + F_{2(m-1)} + F_{m-2}^2 \\
	&amp;= \left( F_{m+2}^2 - F_m^2 \right) + \left( F_m^2 - F_{m-2}^2 \right) + F_{m-2}^2 \\
	&amp;= F_{m+2}^2.
\end{align*}
\end{proof}

Now we derive recursive formulas for $\#(PC_{m,l}^{k})$ and $\#(PC_{m,l}^{k} \oplus S_1)$ in terms of $\#(PC_{m,l}^{k-1})$ and $\#(PC_{m,l}^{k-1} \oplus S_1)$. Remember that $\#(PC_{m,l}^1) = \#(P_m) = L_{2m}$ and $\#(PC_{m,l}^1 \oplus S_1) = \#(P_m \oplus S_1) = F_{2m+2}$. To include all cases together, we extend the Fibonacci sequence in the negative direction by defining $F_{-1}=1$. We also define $X \oplus S_0$ to be $X$ for any finite configuration $X$.  

\begin{figure}[t]
\begin{center}
\resizebox{!}{1.25in}{\includegraphics{P_kml}}
\caption{\scriptsize Using Theorem \ref{thm:loop_algorithm} to compute $\#(PC_{m,l}^{k})$ and $\#(PC_{m,l}^{k} \oplus S_1)$.}
\label{fig:P_kml}
\end{center}
\end{figure}

\begin{theorem} \label{thm:Chain_nums_3} For $m \geq 2$, $k \geq 2$, and $l \geq 1$, 
\[\#(PC_{m,l}^{k}) = \left(L_{2m}F_{l-2} + F_{2m+2}F_{l-1} \right) \#(PC_{m,l}^{k-1}) + \left(L_{2m} F_{l-1} + F_{2m+2}F_{l} \right) \#(PC_{m,l}^{k-1} \oplus S_1)\]
where 
\begin{align*}
\#(PC_{m,l}^{k} \oplus S_1) &amp;= \left(F_{2m+2} F_{l-2} + F_{l-1} F_{m+2}^2 \right) \#(PC_{m,l}^{k-1}) \\
	&amp;\qquad + \left(F_{2m+2} F_{l-1} + F_{l} F_{m+2}^2 \right) \#(PC_{m,l}^{k-1} \oplus S_1).
\end{align*}
\end{theorem}

\begin{proof} We use Theorem \ref{thm:loop_algorithm} applied at the adjacent points indicated with an asterisk as illustrated in Figure \ref{fig:P_kml}, and Corollary \ref{cor:Pm_Sl_plus_St2}:
\begin{align*}
\#(PC_{m,l}^{k}) &amp;= \#(P_m) \#(PC_{m,l}^{k-1} \oplus S_{l-1}) + \#(P_m \oplus S_1) \#(PC_{m,l}^{k-1} \oplus S_{l}) \\
	&amp;= L_{2m} \left(F_{l-1} \#(PC_{m,l}^{k-1} \oplus S_1) + F_{l-2} \#(PC_{m,l}^{k-1}) \right) \\
	&amp; \qquad + F_{2m+2} \left(F_{l} \#(PC_{m,l}^{k-1} \oplus S_1) + F_{l-1} \#(PC_{m,l}^{k-1}) \right) \\
	&amp;= \left(L_{2m} F_{l-1} + F_{2m+2}F_{l} \right) \#(PC_{m,l}^{k-1} \oplus S_1) + \left(L_{2m}F_{l-2} + F_{2m+2}F_{l-1} \right) \#(PC_{m,l}^{k-1}).
\end{align*}

Finally, we determine the recursive formula for $\#(PC_{m,l}^{k} \oplus S_1)$ by using Theorem \ref{thm:loop_algorithm}, applied at the adjacent points indicated with an asterisk in Figure \ref{fig:P_kml}. This gives us 
\[\#(PC_{m,l}^{k} \oplus S_1) = \#(PC_{m,l}^{k-1} \oplus S_{l-1}) \#(P_m \oplus S_1) + \#(PC_{m,l}^{k-1} \oplus S_{l}) \#(P_m \oplus \{S_1[1]; S_1[m+1]\}).\]
Therefore, Lemma \ref{lem:S1_Pm_S1} and Corollary \ref{cor:Pm_Sl_plus_St2} yield 
\begin{align*}
\#(PC_{m,l}^{k} \oplus S_1) &amp;= F_{2m+2} \#(PC_{m,l}^{k-1} \oplus S_{l-1}) + \#(PC_{m,l}^{k-1} \oplus S_{l}) F_{m+2}^2 \\
	&amp;= F_{2m+2} \left(F_{l-1} \#(PC_{m,l}^{k-1} \oplus S_1) + F_{l-2} \#(PC_{m,l}^{k-1}) \right) \\
	&amp; \qquad + \left(F_{l} \#(PC_{m,l}^{k-1} \oplus S_1) + F_{l-1} \#(PC_{m,l}^{k-1}) \right) F_{m+2}^2 \\
	&amp;= \left(F_{2m+2} F_{l-1} + F_{l} F_{m+2}^2 \right) \#(PC_{m,l}^{k-1} \oplus S_1) \\
	&amp; \qquad + \left(F_{2m+2} F_{l-2} + F_{l-1} F_{m+2}^2 \right) \#(PC_{m,l}^{k-1}).
\end{align*}
\end{proof}

As examples of the use of Theorem \ref{thm:Chain_nums_3}, 
\begin{align*}
\#(PC_{3,1}^2) &amp;= L_6 \#(PC_{3,1}^1) + F_{8} \#(PC_{3,1}^1 \oplus S_1) \\
	&amp;= 18 \#(P_3) + 21 \#(P_3 \oplus S_1) \\
	&amp;= 18(18) + 21(21) \\
	&amp;= 765,
\end{align*}
\begin{align*}
\#(PC_{2,2}^2) &amp;= F_{2m+2} \#(PC_{m,l}^{k-1})  +  \left[L_{2m} + F_{2m+2} \right] \#(PC_{m,l}^{k-1} \oplus S_1) \\
	&amp;=  8 \#(P_2) + [7 + 8] \#(P_2 \oplus S_1) \\
	&amp;= (8)(7) + (15)(8)  \\
	&amp;= 176,
\end{align*}
and
\begin{align*}
\#(PC_{2,3}^2) &amp;= \left[L_{4}F_{1} + F_{6}F_{2} \right] \#(PC_{2,3}^{1}) + \left[L_{4} F_{2} + F_{6}F_{l} \right] \#(PC_{2,3}^{1} \oplus S_1)  \\
	&amp;= [(7)(1) + (8)(1)] \#(P_2) + [(7)(1) + (8)(2)] \#(P_2 \oplus S_1)  \\
	&amp;= (15)(7) + (23)(8) \\
	&amp;= 289.
\end{align*}

\section{Polygonal Chain Sequences}

Theorem \ref{thm:Chain_nums_3} gives us recursive formulas for finding $\#(PC_{m,l}^{k})$. By fixing any two of $m, k$, or $l$ and letting the other parameter vary, we create infinite families of integer sequences. Examples of the first few terms of several of these sequences are given in Table \ref{T:Chain_sequences}. We suspect that it is possible to define many new types of configurations in this geometry that provide more interesting and previously unknown integer sequences. 

\begin{table}[t]
\begin{center} 
\begin{tabular}{| c | c | c | c | c | c | c |}  \hline
Sloane A-number &amp; $i$ 		  	&amp;1 		&amp;2 		&amp;3 	 	&amp;4 			&amp;5 \\ \hline 
\seqnum{A152927} &amp; $\#(P_{2,1}^i)$ &amp;7		&amp;113	&amp;1815	&amp;29153		&amp;468263	\\ \hline 
\seqnum{A152928} &amp; $\#(P_{i,1}^2)$	&amp;		&amp;113 	&amp;765	&amp;5234		&amp;35865		 \\ \hline 	
\seqnum{A152929} &amp; $\#(P_{2,i}^2)$	&amp;113	&amp;176	&amp;289	&amp;465		&amp;754		\\ \hline 
\seqnum{A152930} &amp; $\#(P_{2,2}^i)$	&amp;7		&amp;176	&amp;4393	&amp;109649		&amp;2736832	 \\ \hline 	
\seqnum{A152931} &amp; $\#(P_{i,2}^3)$	&amp;		&amp;4393	&amp;80361	&amp;1425131	&amp;25671393 	\\ \hline 
\seqnum{A152932} &amp; $\#(P_{3,i}^3)$	&amp;32733	&amp;80361	&amp;215658	&amp;559305		&amp;1469565  	\\ \hline 
\end{tabular}
\caption{Some polygonal chain sequences}
\label{T:Chain_sequences}
\end{center}
\end{table}

\section{Asymptotic Behavior}

Polygonal chain sequences appear to exhibit asymptotically exponential behavior as the sequences of successive quotients in Table \ref{T:quotients} illustrate. These sequences are natural extensions of string and polygonal sequences, both of which involve Fibonacci and Lucas numbers, so we might expect the ratios of polygonal chain sequences to do so as well. To understand this behavior, we investigate these quotients.

\begin{table}[t]
\begin{center} 
\begin{tabular}{| c | c | c | c | c | c |}  \hline
$i$								&amp;6			&amp;7			&amp;8			&amp;9			&amp;10 \\ \hline
$\frac{\#(P_{2,1}^{i+1})}{\#(P_{2,1}^{i})}$  	&amp;16.06225775	&amp;16.06225775	&amp;16.06225775	&amp;16.06225775	&amp;16.06225775 \\ \hline 
$\frac{\#(P_{i+1,1}^2)}{\#(P_{i,1}^2)}$		&amp;6.854063862	&amp;6.854096407	&amp;6.854101155	&amp;6.854101848	&amp;6.854101949 \\ \hline 	
$\frac{\#(P_{2,i+1}^2)}{\#(P_{2,i}^2)}$		&amp;1.618539787	&amp;1.617840851	&amp;1.618107769	&amp;1.618005808	&amp;1.618044753 \\ \hline 
$\frac{\#(P_{2,2}^{i+1})}{\#(P_{2,2}^i)}$		&amp;24.95993579	&amp;24.95993579	&amp;24.95993579	&amp;24.95993579	&amp;24.95993579 \\ \hline 	
$\frac{\#(P_{i+1,2}^3)}{\#(P_{i,2}^3)}$		&amp;17.95473888	&amp;17.94023899	&amp;17.94580726	&amp;17.94368472	&amp;17.94449609 \\ \hline 
$\frac{\#(P_{3,i+1}^3)}{\#(P_{3,i}^3)}$		&amp;2.619410257	&amp;2.617508525	&amp;2.618234731	&amp;2.617957317	&amp;2.618063275 \\ \hline 
\end{tabular}
\caption{Ratios of polygonal chain sequences to 8 decimal places.}
\label{T:quotients}
\end{center}
\end{table}

\subsection{Fixing $m$ and $l$}
In this section we consider the asymptotic behavior of the sequences obtained by letting $k$ vary for fixed values of $m$ and $l$. For easier notation, let $X_{k,m,l} = \#(PC_{m,l}^k)$ and $Y_{k,m,l} = \#(PC_{m,l}^{k} \oplus S_1)$. Theorem \ref{thm:Chain_nums_3} shows that 

\begin{equation} \label{eq:lim_comb}
X_{k,m,l} = a_{m,l}X_{k-1,m,l} + b_{m,l}Y_{k-1,m,l} \hspace{0.25in} \text{ and } \hspace{0.25in} Y_{k,m,l} = c_{m,l}X_{k-1,m,l} + d_{m,l}Y_{k-1,m,l}
\end{equation}
for values of $a_{m,l},b_{m,l},c_{m,l}$, and $d_{m,l}$ that depend on only $m$ and $l$. We can determine the asymptotic behavior of $X_{k,m,l}$ by writing this system in matrix form:
\[\begin{bmatrix} X_{k,m,l} \\ Y_{k,m,l} \end{bmatrix} = \begin{bmatrix} a_{m,l} &amp; b_{m,l} \\ c_{m,l} &amp; d_{m,l} \end{bmatrix} \ \begin{bmatrix} X_{k-1,m,l} \\ Y_{k-1,m,l} \end{bmatrix}.\]
Let $M_{m,l} = \begin{bmatrix} a_{m,l} &amp; b_{m,l} \\ c_{m,l} &amp; d_{m,l} \end{bmatrix}$. Then 
\[\begin{bmatrix} X_{k,m,l} \\ Y_{k,m,l} \end{bmatrix} =  \left( M_{m,l} \right)^{k-1} \begin{bmatrix} X_{1,m,l} \\ Y_{1,m,l} \end{bmatrix}.\]
The eigenvalues of $M_{m,l}$ are   
\[\lambda_{m,l} = \frac{(d_{m,l}+a_{m,l}) + \sqrt{(d_{m,l}-a_{m,l})^2 + 4b_{m,l}c_{m,l}}}{2}\]
and
\[\overline{\lambda}_{m,l} = \frac{(d_{m,l}+a_{m,l}) - \sqrt{(d_{m,l}-a_{m,l})^2 + 4b_{m,l}c_{m,l}}}{2}.\]
Since $(d_{m,l}-a_{m,l})^2 + 4b_{m,l}c_{m,l} &gt; 0$, we see that $M_{m,l}$ has two distinct real eigenvalues. Thus $M_{m,l}$ is diagonalizable to the matrix $D_{m,l} = \begin{bmatrix} \lambda_{m,l} &amp; 0 \\ 0 &amp; \overline{\lambda}_{m,l} \end{bmatrix}$. If $Q_{m,l}$ is the matrix whose columns are the eigenvectors of $M_{m,l}$, then 
\[\begin{bmatrix} X_{k,m,l} \\ Y_{k,m,l} \end{bmatrix} = \left(Q_{m,l} \begin{bmatrix} \left(\lambda_{m,l}\right)^{k-1} &amp; 0 \\ 0 &amp; \left(\overline{\lambda}_{m,l}\right)^{k-1} \end{bmatrix} Q_{m,l}^{-1} \right) \begin{bmatrix} X_{1,m,l} \\ Y_{1,m,l} \end{bmatrix}.\]
Note that $Q_{m,l} = \begin{bmatrix} 1 &amp; 1 \\ \frac{\lambda_{m,l}-a_{m,l}}{b_{m,l}} &amp; \frac{d_{m,l}- \lambda_{m,l}}{b_{m,l}} \end{bmatrix}$ and $Q_{m,l}^{-1} = \left(\frac{1}{2 \lambda_{m,l} - (a_{m,l}+d_{m,l})}\right) \begin{bmatrix} \lambda_{m,l}-d_{m,l} &amp; b_{m,l} \\ \lambda_{m,l}-a_{m,l} &amp; -b_{m,l} \end{bmatrix}$. 

Since $X_{1,m,l} = L_{2m}$ and $Y_{1,m,l} = F_{2m+2}$, we have  
\begin{equation} \label{eq:recursive}
X_{k,m,l} = R_{m,l}\lambda_{m,l}^{k-1} + S_{m,l} \overline{\lambda}_{m,l}^{k-1}
\end{equation}
where $R_{m,l} = \left(\frac{1}{2\lambda_{m,l}-(a_{m,l}+d_{m,l})} \right) \left( (\lambda_{m,l}-d_{m,l})L_{2m} + b_{m,l}F_{2m+2} \right)$ and \\ $S_{m,l} = \left(\frac{1}{2\lambda_{m,l}-(a_{m,l}+d_{m,l})}\right) \left( (\lambda_{m,l}-a_{m,l})L_{2m} - b_{m,l}F_{2m+2} \right)$. So (\ref{eq:recursive}) provides a non-recursive formula for computing $X_{k,m,l}$. Now $\lambda_{m,l} &gt; \overline{\lambda}_{m,l}$, so we have 
\begin{align*}
\frac{X_{k,m,l}}{X_{k-1,m,l}} &amp;= \frac{R_{m,l}\lambda_{m,l}^{k-1} + S_{m,l} \overline{\lambda}_{m,l}^{k-1}}{R_{m,l}\lambda_{m,l}^{k-2} + S_{m,l} \overline{\lambda}_{m,l}^{k-2}} \\
	&amp;= \frac{R_{m,l} \lambda_{m,l} + S_{m,l} \overline{\lambda}_{m,l}\left(\frac{\overline{\lambda}_{m,l}}{\lambda_{m,l}}\right)^{k-2}}{R_{m,l} + S_{m,l} \left(\frac{\overline{\lambda}_{m,l}}{\lambda_{m,l}}\right)^{k-2}} \\
	&amp;\to \frac{R_{m,l} \lambda_{m,l}}{R_{m,l}} \\
	&amp;= \lambda_{m,l}
\end{align*}
as $k \to \infty$. Thus, 
\begin{equation*} %\label{eq:asymptotic}
\displaystyle \lim_{k \to \infty} \frac{X_{k,m,l}}{X_{k-1,m,l}} = \lambda_{m,l}.
\end{equation*}

The values of $a_{m,l}, b_{m,l}, c_{m,l}$, and $d_{m,l}$ are given by Theorem \ref{thm:Chain_nums_3}: 
\begin{center}
$a_{m,l} = L_{2m}F_{l-2}+F_{2m+2}F_{l-1}$, $b_{m,l} = L_{2m}F_{l-1}+F_{2m+2}F_l,$ \\
$c_{m,l} = F_{2m+2}F_{l-2}+F_{l-1}F_{m+2}^2$, and $d_{m,l} = F_{2m+2}F_{l-1}+F_{l}F_{m+2}^2$.
\end{center}

Thus the sequence of polygonal chain numbers as $k$ varies for fixed $m$ and $l$ behaves asymptotically as the exponential sequence with ratio $\lambda_{m,l}$. For example, $\lambda_{2,1} = 8 + \sqrt{65} \approx 16.06225775$ and $\lambda_{2,2} = \frac{25+3 \sqrt{69}}{2} \approx 24.95993579$ (compare to the ratios in Table \ref{T:quotients}). 

\subsection{Limits of Fibonacci and Lucas Sequences}
To determine the asymptotic behavior of the sequences $\{X_{k,m,l}\}$ as either $m$ or $l$ vary, we will need some limits of sequences of quotients of Fibonacci and Lucas numbers. Let $\varphi = \frac{1 + \sqrt{5}}{2}$ be the golden ratio. The following results concerning Fibonacci and Lucas numbers are well known. 

\begin{itemize}
\item  $\displaystyle \lim_{m \to \infty} \frac{F_{m+1}}{F_m} = \varphi$ (\cite{Koshy} Corollary 8.5, \cite{Lucas} (25))
\item  $\displaystyle \lim_{m \to \infty} \frac{L_{m+1}}{L_m} = \varphi$ (\cite{Koshy} Corollary 8.7, \cite{Lucas} (25))
\item  $\varphi^m = F_m \varphi + F_{m-1}$ for $m \geq 1$ (\cite{Koshy} Lemma 5.1, \cite{BQ} Corollary 33)
\item  $\displaystyle \lim_{m \to \infty} \frac{F_{m+j}}{F_m} = \varphi^j$ for $j \geq 0$ (\cite{BQ} Corollary 31, \cite{Lucas} (27))
\item For $j \geq 0$, $\displaystyle \lim_{m \to \infty} \frac{L_{m+j}}{L_m} = \varphi^j$ (\cite{Lucas} (27))
\end{itemize} 

We will also need other limits of ratios of Fibonacci and Lucas numbers. 

\begin{lemma} \label{lem:seq_ratios} For $m \geq 2$ and $j \geq 0$, 
\begin{enumerate}
\item For $j \geq 0$, $\displaystyle \lim_{m \to \infty} \frac{L_{m}}{F_{m+j}} = \frac{1+\varphi^2}{\varphi^{j+1}}$
\item $\displaystyle \lim_{m \to \infty} \frac{F_{2m+2}}{F_{m+2}^2} = \frac{\varphi^4-1}{\varphi^4} = \frac{1+\varphi^2}{\varphi^3}$
\item $\displaystyle \lim_{m \to \infty} \frac{F_{2m+2}}{F_{m+3}^2} = \frac{\varphi^4-1}{\varphi^6} = \frac{1+\varphi^2}{\varphi^5}$
\item $\displaystyle \lim_{m \to \infty} \frac{F_{2m+4}}{F_{m+2}^2} = \frac{\varphi^4-1}{\varphi^2} = \frac{1+\varphi^2}{\varphi}$
\item $\displaystyle \lim_{m \to \infty} \frac{L_{2m}}{F_{m+2}^2} = \frac{\left(\varphi^2+1\right)^2}{\varphi^6}$
\end{enumerate}
\end{lemma}

\begin{proof} We prove 2 and 5, the remainder are similar. Recalling that $\varphi^2-1 = \varphi$, 
\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{F_{2m+2}}{F_{m+2}^2} &amp;= \displaystyle \lim_{m \to \infty} \frac{F_{m+2}^2 - F_{m}^2}{F_{m+2}^2} \\
	&amp;= \displaystyle \lim_{m \to \infty} 1 - \left(\frac{F_{m}}{F_{m+2}}\right)^2 \\
	&amp;= 1 - \left(\frac{1}{\varphi^2}\right)^2 \\
	&amp;= \frac{\varphi^4 - 1}{\varphi^4} \\
	&amp;= \frac{\left(1+\varphi^2\right)\varphi}{\varphi^4} \\
	&amp;= \frac{1+\varphi^2}{\varphi^3}.
\end{align*}   
Finally, we use Identity 13 from the book \cite{BQ} ($F_{2m+1} = F_m^2 + F_{m+1}^2$) to see 
\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{L_{2m}}{F_{m+2}^2} &amp;= \displaystyle \lim_{m \to \infty} \frac{F_{2m-1} + F_{2m+1}}{F_{m+2}^2} \\
 	&amp;= \displaystyle \lim_{m \to \infty} \frac{(F_{m-1}^2+F_{m}^2) + (F_{m}^2+F_{m+1}^2)}{F_{m+2}^2} \\
	&amp;= \displaystyle \lim_{m \to \infty} \left(\frac{F_{m-1}}{F_{m+2}}\right)^2 + 2\left(\frac{F_{m}}{F_{m+2}}\right)^2 + \left(\frac{F_{m+1}}{F_{m+2}}\right)^2 \\
	&amp;= \left( \frac{1}{\varphi^3} \right)^2 + 2\left( \frac{1}{\varphi^2} \right)^2 + \left(\frac{1}{\varphi}\right)^2 \\
	&amp;= \frac{1 + 2\varphi^2 + \varphi^4}{\varphi^6} \\
	&amp;= \frac{ \left(\varphi^2+1\right)^2}{\varphi^6}.
\end{align*}  
\end{proof}

\subsection{Fixing $k$ and $l$}
In this section, we fix $k$ and $l$ and let $m$ vary and investigate the asymptotic behavior of the resulting sequence $X_{k,m,l}$.  We will need to know limits of ratios of the terms $a_{m,l}, b_{m,l}, c_{m,l}$ and $d_{m,l}$ as $m$ approaches infinity. 

\begin{lemma} \label{lem:m_limits} Let $G_{l} = \frac{ \left(1+\varphi^2\right) F_{l-1} + \varphi^3 F_{l} }{\left(1+\varphi^2\right) F_{l-2} + \varphi^3 F_{l-1}}$. Then for $l \geq 1$, 
\begin{enumerate}
\item $\displaystyle \lim_{m \to \infty} \frac{a_{m+1,l}}{a_{m,l}} = \varphi^2$
\item $\displaystyle \lim_{m \to \infty} \frac{b_{m+1,l}}{a_{m,l}} = \varphi^2 G_{l}$
\item $\displaystyle \lim_{m \to \infty} \frac{b_{m,l}}{a_{m,l}} = G_{l}$
\item $\displaystyle \lim_{m \to \infty} \frac{c_{m,l}}{a_{m,l}} = \frac{\varphi^3}{1+\varphi^2}$
\item $\displaystyle \lim_{m \to \infty} \frac{c_{m+1,l}}{a_{m,l}} = \frac{\varphi^5}{1+\varphi^2}$
\item $\displaystyle \lim_{m \to \infty} \frac{d_{m,l}}{a_{m,l}} = \left(\frac{\varphi^3}{1+\varphi^2} \right) G_{l}.$  
\item $\displaystyle \lim_{m \to \infty} \frac{d_{m+1,l}}{a_{m,l}} = \left(\frac{\varphi^5}{1+\varphi^2} \right) G_{l}.$  
\end{enumerate}
\end{lemma}

\begin{proof} We prove 5, the others are similar: 
\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{c_{m+1,l}}{a_{m,l}} &amp;=\displaystyle \lim_{m \to \infty} \frac{F_{2m+4}F_{l-2}+F_{m+3}^2F_{l-1}}{L_{2m}F_{l-2}+F_{2m+2}F_{l-1}} \\
	&amp;= \lim_{m \to \infty} \frac{\left(\frac{F_{2m+4}}{F_{2m+2}}\right)F_{l-2}+\left(\frac{F_{m+3}^2}{F_{2m+2}}\right)F_{l-1}}{\left(\frac{L_{2m}}{F_{2m+2}}\right)F_{l-2}+F_{l-1}} \\
	&amp;= \frac{ \left( \varphi^2 \right)F_{l-2} + \left( \frac{\varphi^5}{\varphi^2+1} \right)F_{l-1}}{\left(\frac{1+\varphi^2}{\varphi^3}\right)F_{l-2}+F_{l-1}} \\
	&amp;= \left(\frac{\varphi^5}{1+\varphi^2} \right) \left( \frac{ \left(\frac{1+\varphi^2}{\varphi^3}\right)F_{l-2}+F_{l-1} }{ \left(\frac{1+\varphi^2}{\varphi^3}\right)F_{l-2}+F_{l-1} } \right) \\
	&amp;= \frac{\varphi^5}{1+\varphi^2}.
\end{align*}

\end{proof}

Before we can analyze the asymptotic behavior of the polygonal chain sequence, we need to understand how the sequences $\{Y_{k,m,l}\}$ and $\{X_{k,m,l}\}$ are related as $m$ varies. 

\begin{lemma} \label{lem:Y_to_Xm} For $k \geq 1$, 
\[\displaystyle \lim_{m \to \infty} \frac{Y_{k,m,l}}{X_{k,m,l}} = \frac{\varphi^{3}}{1+\varphi^2}.\]
\end{lemma}

\begin{proof} When $k=1$ we have $\displaystyle \lim_{m \to \infty} \frac{Y_{1,m,l}}{X_{1,m,l}} = \lim_{m \to \infty} \frac{F_{2m+2}}{L_{2m}} = \frac{1}{\frac{1}{\varphi^3} + \frac{1}{\varphi}} =  \frac{\varphi^3}{1+\varphi^2}$. So the claim is true for $k = 1$. We proceed by induction and assume the claim is true for some $k \geq 1$. Then (\ref{eq:lim_comb}) shows 

\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{Y_{k+1,m,l}}{X_{k+1,m,l}} &amp;= \lim_{m \to \infty} \frac{c_{m,l} X_{k,m,l} + d_{m,l}Y_{k,m,l}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{m \to \infty} \frac{ \left(\frac{c_{m,l}}{a_{m,l}}\right) + \left(\frac{d_{m,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)}{1 + \left(\frac{b_{m,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)} \\
	&amp;= \frac{ \left( \frac{\varphi^3}{1+\varphi^2}\right) + \left( \frac{ \varphi^3}{1+\varphi^2} \right)G_{l} \left( \frac{\varphi^3}{1+\varphi^2} \right)}{1+ G_{l}\left( \frac{\varphi^3}{\varphi^2+1} \right)} \\
	&amp;= \frac{\varphi^3}{1+\varphi^2}.
\end{align*}  
\end{proof}

Now we can identify the asymptotic behavior of the polygonal chain sequence when $m$ varies. 

\begin{theorem} For $k \geq1$ and $l \geq 1$, 
\begin{equation*} \label{eq:seqm}
\displaystyle \lim_{m \to \infty} \frac{X_{k,m+1,l}}{X_{k,m,l}} = \varphi^{2k} \text{ and } \displaystyle \lim_{m \to \infty} \frac{Y_{k,m+1,l}}{X_{k,m,l}} = \frac{\varphi^{2k+3}}{1+\varphi^2}.
\end{equation*}
\end{theorem}

\begin{proof} The proof will be by induction on $k$. Now $X_{1,m,l} = L_{2m}$ and $\displaystyle \lim_{m \to \infty} \frac{X_{1,m+1,l}}{X_{1,m,l}} = \lim_{m \to \infty} \frac{L_{2m+2}}{L_{2m}} = \varphi^2$. Also, $Y_{1,m,l} = F_{2m+2}$ and $\displaystyle \lim_{m \to \infty} \frac{Y_{1,m+1,l}}{X_{1,m,l}} = \lim_{m \to \infty} \frac{F_{2m+4}}{L_{2m}} = \frac{1}{\frac{1}{\varphi^5}+\frac{1}{\varphi^3}} = \frac{\varphi^5}{1+\varphi^2}$. Thus, our theorem is true when $k=1$. Assume the theorem is true for some $k \geq 1$. 

Then 
\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{X_{k+1,m+1,l}}{X_{k+1,m,l}} &amp;= \lim_{m \to \infty} \frac{a_{m+1,l} X_{k,m+1,l} + b_{m+1,l}Y_{k,m+1,l}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{m \to \infty} \frac{ \left(\frac{a_{m+1,l}}{a_{m,1}}\right)\left(\frac{X_{k,m+1,l}}{X_{k,m,l}}\right) + \left(\frac{b_{m+1,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m+1,l}}{X_{k,m,l}}\right)}{1 + \left(\frac{b_{m,l}}{a_{m,1}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)} \\
	&amp;= \frac{ \left(\varphi^2\right)\left(\varphi^{2k}\right) + \left( \varphi^2 G_{l} \right) \left(\frac{\varphi^{2k+3}}{1+\varphi^2}\right)}{1 +  \left( G_{l} \right)\left(\frac{\varphi^3}{1+\varphi^2}\right)} \\
	&amp;= \varphi^{2k+2}
\end{align*}  

and

\begin{align*}
\displaystyle \lim_{m \to \infty} \frac{Y_{k+1,m+1,l}}{X_{k+1,m,l}} &amp;= \lim_{m \to \infty} \frac{c_{m+1,l} X_{k,m+1,l} + d_{m+1,l}Y_{k,m+1,l}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{m \to \infty} \frac{ \left(\frac{c_{m+1,l}}{a_{m,l}}\right)\left(\frac{X_{k,m+1,l}}{X_{k,m,l}}\right) + \left(\frac{d_{m+1,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m+1,l}}{X_{k,m,l}}\right)}{1 + \left(\frac{b_{m,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)} \\
	&amp;= \frac{ \left( \frac{\varphi^5}{1+\varphi^2} \right) \left(\varphi^{2k}\right) + \left(\frac{\varphi^5}{1+\varphi^2} \right)G_{l} \left(\frac{\varphi^{2k+3}}{1+\varphi^2}\right)}{ 1 + \left( G_{l} \right) \left(\frac{\varphi^3}{1+\varphi^2}\right) } \\
	&amp;= \frac{\varphi^{2k+5}}{1+\varphi^2}.
\end{align*}  

\end{proof}

So the sequence of polygonal chain numbers as $m$ varies for fixed $k$ and $l$ behaves asymptotically as the exponential sequence with ratio $\varphi^{2k}$. For example, $\varphi^{4} \approx 6.854101954$ and $\varphi^{6} \approx 17.94427186$ (compare to the ratios in Table \ref{T:quotients}). 
Note that the limits in these cases are independent of the value of $l$. 

\subsection{Fixing $k$ and $m$}
In this section, we fix $k$ and $m$ and let $l$ vary and investigate the asymptotic behavior of the resulting sequence $X_{k,m,l}$. Again, we begin with limits of sequences of ratios of $a_{m,l}, b_{m,l}, c_{m,l}$, and $d_{m,l}$, this time as $l$ goes to infinity. 

\begin{lemma} \label{lem:coeffseq_l} Let $H_m = \frac{F_{2m+2}+F_{m+2}^2 \varphi }{L_{2m}+F_{2m+2} \varphi}$. Then for $m \geq 2$,
\begin{enumerate}
\item $\displaystyle \lim_{l \to \infty} \frac{a_{m,l+1}}{a_{m,l}} = \varphi$
\item $\displaystyle \lim_{l \to \infty} \frac{b_{m,l}}{a_{m,l}} = \varphi$
\item $\displaystyle \lim_{l \to \infty} \frac{b_{m,l+1}}{a_{m,l}} = \varphi^2 $
\item $\displaystyle \lim_{l \to \infty} \frac{c_{m,l}}{a_{m,l}} = H_m$
\item $\displaystyle \lim_{l \to \infty} \frac{c_{m,l+1}}{a_{m,l}} = \left( \varphi \right) H_m$
\item $\displaystyle \lim_{l \to \infty} \frac{d_{m,l+1}}{a_{m,l}} = \left( \varphi^2 \right) H_m$.
\end{enumerate}
\end{lemma}

\begin{proof} We verify 5 and leave the others for the reader:
\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{c_{m,l+1}}{a_{m,l}} &amp;= \displaystyle \lim_{l \to \infty} \frac{F_{2m+2}F_{l-1}+F_{m+2}^2F_{l}}{L_{2m}F_{l-2}+F_{2m+2}F_{l-1}} \\
	&amp;= \lim_{l \to \infty} \frac{ F_{2m+2} + F_{m+2}^2 \left(\frac{F_{l}}{F_{l-1}}\right)}{L_{2m} \left(\frac{F_{l-2}}{F_{l-1}}\right) + F_{2m+2}} \\
	&amp;= \frac{F_{2m+2} + F_{m+2}^2 \varphi}{L_{2m} \left( \frac{1}{\varphi} \right) + F_{2m+2}} \\
	&amp;= \left( \varphi \right) \left( \frac{F_{2m+2}+F_{m+2}^2 \varphi }{L_{2m}+F_{2m+2} \varphi} \right).
\end{align*}
\end{proof}

Now we need to understand the relationship between the sequence $\{Y_{k,m,l}\}$ and the sequence $\{X_{k,m,l}\}$ as $l$ goes to infinity.

\begin{lemma} \label{lem:Y_to_X_l} For $k \geq 2$, $m \geq 2$,  
\[\displaystyle \lim_{l \to \infty} \frac{Y_{k,m,l}}{X_{k,m,l}} = H_m.\]
\end{lemma}

\begin{proof} When $k=2$ we have 
\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{Y_{2,m,l}}{X_{2,m,l}} &amp;= \lim_{l \to \infty} \frac{c_{m,l}X_{1,m,l} + d_{m,l}Y_{1,m,l}}{a_{m,l}X_{1,m,l} + b_{m,l}Y_{1,m,l}} \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{c_{m,l}}{a_{m,l}}\right) X_{1,m,l} + \left(\frac{d_{m,l}}{a_{m,l}}\right) Y_{1,m,l} }{X_{1,m,l} + \left(\frac{b_{m,l}}{a_{m,l}}\right) Y_{1,m,l} } \\
	&amp;= \frac{ \left( H_m \right) L_{2m} + \left( \varphi H_m \right) F_{2m+2}}{ L_{2m} + \varphi F_{2m+2}} \\
	&amp;= H_m.	
\end{align*}
	
So the lemma is true for $k = 2$. We proceed by induction and assume the lemma is true for some $k \geq 2$. Then (\ref{eq:lim_comb}) shows 
\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{Y_{k+1,m,l+1}}{X_{k+1,m,l+1}} &amp;= \lim_{l \to \infty} \frac{c_{m,l} X_{k,m,l} + d_{m,l}Y_{k,m,l}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{c_{m,l}}{a_{m,l}}\right) + \left(\frac{d_{m,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)}{1 + \left(\frac{b_{m,l}}{a_{m,l}}\right)\left(\frac{Y_{k,m,l}}{X_{k,m,l}}\right)} \\
	&amp;=  \frac{ \left( H_m  \right) + \left( \varphi H_m \right) \left( H_m \right) }{1 + \left( \varphi \right)\left( H_m \right)} \\
	&amp;= H_m.
\end{align*}  
\end{proof}

Now we determine the asymptotic behavior of the polygonal chain sequences if we allow $l$ to vary.

\begin{theorem} For $k \geq 2$ and $m \geq 2$, 
\begin{equation*} \label{eq:seql}
\displaystyle \lim_{l \to \infty} \frac{X_{k,m,l+1}}{X_{k,m,l}} = \varphi^{k-1} \text{ and } \displaystyle \lim_{l \to \infty} \frac{Y_{k,m,l+1}}{X_{k,m,l}} = \varphi^{k-1} \left( H_m \right).
\end{equation*}
\end{theorem}

\begin{proof} The proof will be by induction on $k$. Recall that $X_{1,m,l} = L_{2m}$ and $Y_{1,m,l} = F_{2m+2}$ We first verify the $k=2$ case. Equation (\ref{eq:lim_comb}) shows 
\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{X_{2,m,l+1}}{X_{2,m,l}} &amp;= \lim_{l \to \infty} \frac{a_{m,l+1} X_{1,m,l+1} + b_{m,l+1}Y_{1,m,l+1}}{a_{m,l} X_{1,m,l} + b_{m,l}Y_{1,m,l}}  \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{a_{m,l+1}}{a_{m,l}}\right) X_{1,m,l+1} + \left(\frac{b_{m,l+1}}{a_{m,l}}\right) Y_{1,m,l+1} }{X_{1,m,l} + \left(\frac{b_{m,l}}{a_{m,l}}\right) Y_{1,m,l} } \\
	&amp;= \frac{ \varphi L_{2m} + \varphi^2 F_{2m+2}}{ L_{2m} + \varphi F_{2m+2}} \\
	&amp;= \varphi
\end{align*}  

and

\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{Y_{2,m,l+1}}{X_{2,m,l}} &amp;= \lim_{l \to \infty} \frac{c_{m,l+1} X_{1,m,l+1} + d_{m,l+1}Y_{1,m,l+1}}{a_{m,l} X_{1,m,l} + b_{m,l}Y_{1,m,l}}  \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{c_{m,l+1}}{a_{m,l}}\right) X_{1,m,l+1} + \left(\frac{d_{m,l+1}}{a_{m,l}}\right) Y_{1,m,l+1} }{X_{1,m,l} + \left(\frac{b_{m,l}}{a_{m,l}}\right) Y_{1,m,l} } \\
	&amp;= \frac{ \left( \varphi \right) \left( H_m \right) L_{2m} + \left( \varphi^2 H_m \right) F_{2m+2}}{ L_{2m} + \varphi F_{2m+2}} \\
	&amp;= \varphi H_m.
\end{align*}

Now the general cases: 

\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{X_{k+1,m,l+1}}{X_{k+1,m,l}} &amp;= \lim_{l \to \infty} \frac{a_{m,l+1} X_{k,m,l+1} + b_{m,l+1}Y_{k,m,l+1}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{a_{m,l+1}}{a_{m,l}}\right) \left( \frac{X_{k,m,l+1}}{X_{k,m,l}} \right) + \left(\frac{b_{m,l+1}}{a_{m,l}}\right) \left( \frac{Y_{k,m,l+1}}{X_{k,m,l}} \right) }{1 + \left(\frac{b_{m,l}}{a_{m,l}}\right) \left( \frac{Y_{k,m,l}}{X_{k,m,l}} \right) } \\
	&amp;= \frac{ \left( \varphi \right) \left( \varphi^{k-1} \right) + \left( \varphi^2 \right) \left( \varphi^{k-1} H_m \right)}{ 1 + \left( \varphi \right) \left( H_m \right) } \\
	&amp;= \varphi^k
\end{align*}  

and

\begin{align*}
\displaystyle \lim_{l \to \infty} \frac{Y_{k+1,m,l+1}}{X_{k+1,m,l}} &amp;= \lim_{l \to \infty} \frac{c_{m,l+1} X_{k,m,l+1} + d_{m,l+1}Y_{k,m,l+1}}{a_{m,l} X_{k,m,l} + b_{m,l}Y_{k,m,l}}  \\
	&amp;= \displaystyle \lim_{l \to \infty} \frac{ \left(\frac{c_{m,l+1}}{a_{m,l}}\right) \left( \frac{X_{k,m,l+1}}{X_{k,m,l}} \right) + \left(\frac{d_{m,l+1}}{a_{m,l}}\right) \left( \frac{Y_{k,m,l+1}}{X_{k,m,l}} \right) }{1 + \left(\frac{b_{m,l}}{a_{m,l}}\right) \left( \frac{Y_{k,m,l}}{X_{k,m,l}} \right) } \\
	&amp;= \frac{ \left( \varphi H_m \right) \left( \varphi^{k-1} \right) +  \left( \varphi^2 H_m \right) \left( \varphi^{k-1} H_m \right)}{ 1 +   \left( \varphi \right) \left( H_m \right) } \\
	&amp;= \varphi^{k} H_m.
\end{align*}  

\end{proof}

So the sequence of polygonal chain numbers as $l$ varies for fixed $k$ and $m$ behaves asymptotically as the exponential sequence with ratio $\varphi^{k-1}$. For example, $\varphi \approx 1.618033988$ and $\varphi^{2} \approx 2.618033986$ (compare to the ratios in Table \ref{T:quotients}). 
Note that the limits in these cases are independent of the value of $m$. 

\section{Conclusion}
The asymptotic growth rates of polygonal chain sequences are functions of the golden ratio $\varphi$. Since polygonal chains are constructed from string and polygonal configurations and $\displaystyle \lim_{l \to \infty} \frac{\#(S_{l+1})}{\#(S_l)} = \varphi$ and $\displaystyle \lim_{m \to \infty} \frac{\#(P_{m+1})}{\#(P_m)} = \varphi^2$, this should not be surprising in hindsight. That the asymptotic ratios of the polygonal chain sequences have such nice closed forms is, however, somewhat surprising given the complex nature of betweenness in this geometry. 

Polygonal chain sequences provide a three parameter family of integer sequences previously unidentified in The On-Line Encyclopedia of Integer Sequences \cite{OEIS}. These sequences have interesting geometric interpretations as the number of compact sets at each location between the component sets that make up the polygonal chain. We suspect that there are many other similar families of unknown sequences just waiting to be found. 

\section{Acknowledgements}
This work was supported by National Science Foundation grant DMS-0451254. The authors also thank Grand Valley State University for hosting and supporting a Research Experiences for Undergraduates program. 

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\bibitem{OEIS}
N. J. A. Sloane, (2007), The On-Line Encyclopedia of Integer Sequences,
published electronically at \href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/\char'176njas/sequences/}

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\noindent 2000 {\it Mathematics Subject Classification}: 
Primary 51F99; Secondary 11B83.

\noindent \emph{Keywords:}  Hausdorff metric, configuration, metric geometry, polygonal chains, Fibonacci numbers, Lucas numbers.

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\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A152927},
\seqnum{A152928},
\seqnum{A152929},
\seqnum{A152930},
\seqnum{A152931}, and
\seqnum{A152932}.)

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\noindent
Received August 22 2008;
revised version received December 22 2008.  
Published in {\it Journal of Integer Sequences}, December 22 2008.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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