<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}
\usepackage{amsthm}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}

\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\begin{center}
\vskip 1cm{\LARGE\bf 
Generalizing the Conway-Hofstadter \$10,000 Sequence}
\vskip 1cm
\large
John A.\ Pelesko\\
Department of Mathematical Sciences\\
University of Delaware \\
Newark, DE 19716\\
USA \\
\href{mailto:pelesko@math.udel.edu}{\tt pelesko@math.udel.edu} \\
\end{center}

\vskip .2 in
\begin{abstract}
We introduce a generalization of the Conway-Hofstadter \$10,000 sequence. The
sequences introduced, called \emph{k-sequences}, preserve the
Conway-Hofstadter-Fibonacci-like structure of forming terms in the sequence by
adding together two previous terms, equidistant from the start and end of the
sequence. We examine some particular $k$-sequences, investigate relationships to
known integer sequences, establish some properties which hold for all $k$, and
show how to solve many of the defining nonlinear recursions by examining related
underlying sequences termed \emph{clock} sequences.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section] 

%\documentclass[12pt]{article}
%\usepackage{amssymb,amsthm,amssymb}
%\usepackage[first,bottomafter]{draftcopy}
%\usepackage{epsfig}

%\renewcommand{\baselinestretch}{1.2}
%This is the command that spaces the manuscript for easy reading

%These are abbreviations of J.A. Pelesko
\newcommand{\eqnref}[1]{\ref{eq:#1}}
\newcommand{\secref}[1]{section~\ref{sec:#1}}
\newcommand{\figref}[1]{Figure~\ref{fig:#1}}
\newcommand{\tabref}[1]{Table~\ref{tab:#1}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ra}{\rightarrow}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\dt}{\delta t}
\newcommand{\beas}{\begin{eqnarray*}}
\newcommand{\eeas}{\end{eqnarray*}}
\newcommand{\xh}{\hat{x}}
\newcommand{\expand}[2]{#1(#2) \sim #1_0(#2) + \epsilon #1_1(#2) + \epsilon^2 #1_2(#2) + \cdots}
\newcommand{\R}{\mathbf{R}}
\newcommand{\norm}[1]{|| #1 ||_{\infty}}
%\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{prop}{Proposition}
\newcommand\around[1]{\ensuremath{\mathbin{\settowidth{\dimen7}{\mbox{$\bigcirc$}}%
              \makebox[0pt][l]{$\bigcirc$}\makebox[\dimen7]{#1}}}}


%\begin{document}
%\thispagestyle{empty}




\section{Introduction}
In a talk at AT\&amp;T Bell Labs \cite{conwaytalk} in 1988, J.H.\ Conway introduced
the sequence (A004001 in the On-Line Encyclopedia of Integer Sequences) \beas
1,1,2,2,3,4,4,4,5,6,7,7,8,8,8,8,9,\ldots \eeas defined by the recursion \be
c(n)=c(c(n-1))+c(n-c(n-1)), \label{eq:c1} \ee with initial conditions
$c(1)=c(2)=1$. Conway had proven that $c(n)/n \rightarrow 1/2$, but was unable
to establish the rate of convergence. Somewhat overestimating the difficulty of
the question he offered a prize of \$10,000 to the first person who could. The
challenge was answered by C.L.\ Mallows \cite{Mallows} shortly thereafter.
Mallows not only established the rate of convergence, but uncovered additional
structure in the sequence as well. The exchange caught the attention of the
popular press inspiring an entertaining article in the {\it New York Times}
\cite{NYT}. The popularization of A004001 generated by this exchange also led to
the study of Kubo and Vakil \cite{Kubo} where much of the combinatorial
structure of the sequence was unveiled. The use of a \emph{compression
operation} to characterize the sequence allowed for simple proofs of many of
A004001's interesting properties. We also note that unbeknownst to Conway and
Mallows the sequence had previously been introduced by Hofstadter
\cite{Hofstadter} and had also appeared in the problems section of the American
Mathematical Monthly \cite{Newman}. Today A004001 is known either as the
``Conway-Hofstadter \$10,000 sequence'' or as the ``Conway-Newman'' sequence.
We will refer to it as the ``Conway-Hofstadter'' sequence.

Many of the properties that inspired interest in (\ref{eq:c1}) are nicely
enumerated by Kubo and Vakil \cite{Kubo}. For convenience of the reader we list
those relevant to this paper here:
\begin{enumerate}

\item $c(n) \leq n$.

\item $c(n)-c(n-1) = 0$ or $1$, for all $n \geq 1$.

\item $c(n) \geq n/2$, with equality iff $n$ is a power of $2$ and $n \neq 1$.

\item $c(n)/n \longrightarrow 1/2$ as $n \longrightarrow \infty$.

\item $c(2n) \leq 2 c(n)$ for all $n$.

\end{enumerate}

In this paper we generalize the Conway-Hofstadter sequence.
Our generalized class of sequences shows
much of the structure of (\ref{eq:c1}), but also exhibits interesting new
behavior. The generalization leads to new representations of old sequences and
to new solvable nonlinear recursions.

\section{The Generalization: \emph{k-Sequences}}
In reading the work of Mallows \cite{Mallows} or Kubo and Vakil \cite{Kubo} one
is immediately struck by the statements following the presentation of the first
property listed above of the Conway-Hofstadter sequence. Both authors note that $c(n) \leq n$ and then
go on to say ``so that $c(n)$ is well-defined by the recurrence.'' To understand
this comment and to appreciate the motivation for our generalization it is worth
visualizing how terms in the Conway-Hofstadter sequence are formed. Consider the first five terms of
the Conway-Hofstadter sequence: \beas 1,1,2,2,3. \eeas To form the sixth term, we note that the fifth
term is equal to $3$, count forward from the beginning of the sequence three
terms, backwards from the end of the sequence three terms, and add the results
to find $c(6)=2+2=4$. This procedure generates all terms in the sequence. Note
that there is a beautiful symmetry in this construction process; in forming the
$n$th term, one term from the first half of the sequence is added to a term from
the second half of the sequence. These terms are always equidistant from the
start and end of the sequence. The observation of Mallows or Kubo and Vakil is
equivalent to noting that $c(n) \leq n$ assures that we never count past the end
(or the beginning) of the sequence. Of course, if we consider clock or modular
arithmetic, counting beyond the start or end of the sequence is no longer a
problem. This immediately suggests our generalization.
\begin{definition}
We say that $\{c_k(n)\}$ is a Conway-Hofstadter-like sequence of order $k$ when
defined by the recursion \be c_k(n) = c_k( k c_k(n-1) \bmod (n-1)) + c_k(n-k
c_k(n-1) \bmod (n-1)) \ee with $c_k(1)=c_k(2)=1$. We also call such sequences,
k-sequences, and denote them $c_k$.
\end{definition}
Note that this generalization preserves the symmetry of the Conway-Hofstadter sequence. That is, the
only modifications to the construction process above are that the last term in
the sequence in multiplied by $k$, and the count from the start and end of the
sequence is done using modular arithmetic.
However, in forming the $n$th term,
one term from the first half of the sequence is still added to one term from the
second half of the sequence; these terms are again equidistant from the start
and end of the sequence. Throughout this paper, we observe the
convention that a zero in modular arithmetic mod $n$ is replaced by $n$.


\subsection{A Glance at Some $k$-Sequences} \label{sec:obs}
Computing the first few terms of $k$-sequences for various $k$ reveals some
familiar sequences hiding among the $k$'s as well as some new surprises. The
observed behavior of the first fifteen $k$-sequences is summarized in the
following table:

\begin{table}[H]
\begin{center}
\begin{tabular}{|c|l|} \hline
$k$-sequence &amp;  First twenty terms \\ \hline \hline
$c_1$ &amp; 1,1,2,2,3,4,4,4,5,6,7,7,8,8,8,8,9,10,11,12 \ldots \\ \hline
$c_2$ &amp; 1,1,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11 \ldots \\ \hline
$c_3$ &amp; 1,1,2,3,4,4,5,6,6,7,8,8,9,10,10,11,12,12,13,14 \ldots \\ \hline
$c_4$ &amp; 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10 \ldots \\ \hline
$c_5$ &amp; 1,1,2,3,3,4,4,5,6,6,7,7,8,9,9,10,10,11,12,12 \ldots \\ \hline
$c_6$ &amp; 1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,11,12,13,13 \ldots \\ \hline
$c_7$ &amp; 1,1,2,2,3,4,4,5,6,6,7,7,8,8,9,10,11,12,11,12 \ldots \\ \hline
$c_8$ &amp; 1,1,2,3,4,4,5,6,7,7,8,9,10,10,11,12,13,13,14,15 \ldots \\ \hline
$c_9$ &amp; 1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,11,12,13,13 \ldots \\ \hline
$c_{10}$ &amp; 1,1,2,2,3,4,4,5,5,6,7,7,8,8,9,10,10,11,11,12 \ldots \\ \hline
$c_{11}$ &amp; 1,1,2,3,4,4,5,5,6,7,8,9,9,9,10,12,12,13,13,14 \ldots \\ \hline
$c_{12}$ &amp; 1,1,2,3,4,4,5,6,7,7,8,9,10,10,11,12,13,13,14,15 \ldots \\ \hline
$c_{13}$ &amp; 1,1,2,2,3,3,4,5,6,5,6,7,7,8,9,9,10,10,11,10 \ldots \\ \hline
$c_{14}$ &amp; 1,1,2,3,3,4,4,5,6,6,7,7,8,9,10,10,10,12,12,13 \ldots \\ \hline
$c_{15}$ &amp; 1,1,2,3,4,5,5,6,6,7,9,8,9,10,11,12,12,13,14,15 \ldots \\ \hline
\end{tabular}
\end{center}
\end{table}

The sequence $c_1$ is, of course, the Conway-Hofstadter sequence.
Other familiar sequences are lurking in this list. The sequence $c_4$
is the nice sequence $\lfloor (n+1)/2 \rfloor$ which is equivalent to A004526.
Both $c_6$ and $c_9$ appear to follow the pattern ``one even followed by two
odd'' and hence appear equivalent to A004396. The sequences $c_8$ and $c_{12}$
appear equivalent to A037915, or more simply $\lfloor (3n+4)/4 \rfloor$.
These suggested equivalences require proof.
We do not present such
proof at this point. Rather, we will first establish some general
results about the $c_k$'s, then, demonstrate a method for uncovering the
hidden structure of many $c_k$'s, and finally show how to prove an
equivalence suggested by the table above. We do at this point note that
direct computation of various $k$-sequences highlights interesting
similarities and differences between $c_1$ and other $c_k$'s.
Properties (1) and (3) of $c_1$ appear to be satisfied for all
$k$. Property (2) however is violated for most $c_k$'s. This is seen
both through failure of monotonicity and in the
``skipping'' of integers in sequences such as $c_{11}$ and
$c_{13}$. Of course, the other striking feature of the $c_k$ is
the apparent new representation of some familiar sequences such as
A004526 or A004396.

\section{Properties of the $c_k$}
Observations suggest that all $c_k$ satisfy upper and lower bounds on growth
similar to the bounds on $c_1$. This is indeed true and we have
\begin{prop}
$c_k(n) \leq n$ for all $n \geq 1$, $k \geq 1$.
\end{prop}
\begin{proof} For any fixed $k$, we proceed by induction on $n$. Note that
$c_k(1)=c_k(2)=1$ and $c_k(3)=2$ and hence $c_k(n) \leq n$ for $1
\leq n \leq 3$. Now, assume $c_k(j) \leq j$ for all $j$ satisfying
$1 \leq j \leq n$ and consider $c_k(n+1)$. We have \beas c_k(n+1)
= c_k(k c_k(n) \bmod n) + c_k(n+1 - k c_k(n) \bmod n).\eeas Let
$j=k c_k(n) \bmod n$ and observe that $1 \leq j \leq n$. Hence
\beas c_k(n+1) = c_k(j) + c_k(n+1-j) \leq j+n+1-j = n+1 \eeas and
\beas c_k(n+1) \leq n+1 \eeas as desired.\end{proof} A similar
argument yields the lower bound
\begin{prop}
$c_k(n) \geq n/2$ for all $n \geq 1$, $k \geq 1$.
\end{prop}
A key difference between $c_1$ and a general $c_k$ is that property (2) need not
hold. This allows a particular $c_k$ to be non-monotone and to ``skip''
integers. For example, $c_7(19)-c_7(18)=-1$ demonstrating the non-monotone
property while $c_{11}$ does not contain the number $11$, as we shall
soon see. The bounds
above immediately provide a means to prove that integers can indeed be skipped. We have
\begin{prop}
Let $m&gt;0$ and suppose $m$ does not appear in the first $N$ terms of $c_k(n)$
where $N &gt; 2m$, then $m$ never appears in $c_k(n)$.
\end{prop}
\begin{proof} $ c_k(N) \geq \frac{N}{2} &gt; m. $ \end{proof}
Notice that this proposition, along with computation of the first $23$ terms of
$c_{11}$ establishes that $c_{11}$ is indeed ``missing'' $11$. We can also
easily bound the maximum number of occurrences of a particular integer in the
sequence $c_k$.
\begin{prop} Let $f_k(m)$ denote the total number of occurrences of $m$ in the
sequence $c_k$. Then, $f_k(m) \leq m+1$ for all $k$ and $m$.
\end{prop}
\begin{proof}
By our lower bound on $c_k$ we have $c_k(2m) \geq m$. By our upper bound we have
$c_k(m) \leq m$. Hence $m$ can only appear amongst the $m+1$ terms $c_m,
c_{m+1},\ldots c_{2m}$.
\end{proof}

Another natural question is whether or not $c_k$ and $c_j$ can be
``equivalent.'' We consider two notions of equivalence.
\begin{definition}
We say that $c_k$ and $c_j$ are numerically equivalent to order $N$ iff
$c_k(n)=c_j(n)$ for all $n$ satisfying $1 \leq n \leq N$. We say that $c_k$ and
$c_j$ are structurally equivalent to order $N$ iff $k c_k(n) \bmod n = j c_j(n)
\bmod n$ for all $n$ satisfying $3 \leq n \leq N$.
\end{definition}
Structural equivalence tracks the process of forming a
$k$-sequence. It decides whether or not two $k$-sequences were
formed by adding together terms located at the same point in each
sequence. Structural equivalence clearly implies numerical
equivalence. The converse is not true. It is possible for two
sequences to be numerically equivalent, but not structurally
equivalent. Of the two types of equivalence, we consider
structural equivalence to be fundamental. We can compute the set
of all $k$-sequences structurally equivalent to a given sequence,
$c_j$, by solving a system of linear congruences. For example,
consider $c_2$, which begins $\{1,1\}$. Since $2 \equiv 0$ (mod
$2$) we may find all $k$-sequences structurally equivalent to
order $2$ by solving the congruence $k \equiv 0$ (mod $2$). The
set of even integers satisfies this congruence. At the next step,
$c_2$ is $\{1,1,2\}$, and since $4 \equiv 1$ (mod $3$) we must
solve the congruence $2k \equiv 1$ (mod $3$). The solutions to
this are numbers in the arithmetic progression $2,5,8,\ldots$.
Hence the $k$-sequences structurally equivalent to $c_2$ at order
$3$ are those in the progression with steps of length $2 \times
3$, i.e., $k = 2,8,14,\ldots$. Replacing $2$ with $j$ and
generalizing the argument above we may show
\begin{prop}
Given any $k,N$, there exists a $j \neq k$, such that $c_k$ and $c_j$ are structurally
equivalent to order $N$. Further, if $j$ is the smallest such integer, $j \rightarrow \infty$
as $N \rightarrow \infty$.
\end{prop}
Notice that this proposition implies that no two $k$-sequences are
structurally equivalent of infinite order. In this sense, the
$k$-sequences are distinct. As mentioned above, two $k$-sequences
may be numerically equivalent, but not structurally equivalent.
Numerical investigation suggests that numerical equivalence is
infrequent.

\section{The Beat of the $c_k$'s}
The structure implicit in the notion of structural equivalence can also shed
light on the behavior of particular $k$-sequences. The underlying structure of
a given $k$-sequence, that is the sequence of terms used to create $c_k$, is
tracked by the associated \emph{clock} sequence.
\begin{definition}
Associated with each $k$-sequence, $c_k$, we define a clock sequence, denoted
$t_k$, as the sequence satisfying
\beas
t_k(n) = \min (kc_k(n-1) \bmod (n-1), n-k c_k(n-1) \bmod(n-1))
\eeas
for $n \geq 3$ with $t_k(1)=t_k(2)=1$.
\end{definition}
Note that the clock, $t_k(n)$, starting at $n=3$, tracks the
term from the \emph{lower} half of the sequence of length
$n-1$ that is used to compute the $n$th term.
In terms of its clock, a $k$-sequence can be written
\beas
c_k(n) = c_k(t_k(n-1)) + c_k(n-t_k(n-1)),
\eeas
again where $n \geq 3$.
A clock sequence becomes particularly useful when it become \emph{periodic}.
For example, consider the growth of $c_2$. We circle the terms at level $n-1$ used
to create the new term at level $n$:
\begin{center}
\begin{tabular}{|c|} \hline
\around{$1$}, \around{$1$} \\

\around{$1$},$1$,\around{$2$} \\

$1$,\around{$1$},\around{$2$},$3$ \\

\around{$1$},$1$,$2$,$3$,\around{$3$} \\

$1$,\around{$1$},$2$,$3$,\around{$3$},$4$ \\

\around{$1$},$1$,$2$,$3$,$3$,$4$,\around{$4$} \\

$1$,\around{$1$},$2$,$3$,$3$,$4$,\around{$4$},$5$ \\ \hline
\end{tabular}
\end{center}
The regular visual pattern translates into periodic behavior of
$t_2$. In particular, $t_2 = 1,1,1,1,2,1,2,1,2,1,2,\ldots$. If we
conjecture that this pattern continues, we may extract from $t_2$
the simpler set of \emph{linear} recursions satisfied by $c_2$ \be
c_2(2n) = 1 + c_2(2n-1), \ee \be c_2(2n+1) = 1 + c_2(2n-1), \ee
from which the description of $c_2$ in our table and properties
such as (4) and (5) of $c_1$ may easily be established. One might
conjecture (wishfully) that a clock sequence that repeats itself
continues to do so. Unfortunately, the clock of $c_7$ already
furnishes a counterexample, repeating a portion of itself, and
then wandering off into apparent aperiodic behavior. However, when
a $c_k$ does behavior in a regular, if perhaps complicated,
fashion, clock sequences allow us to uncover this hidden
structure. To search for this structure in a given $c_k$, we can
plot the phase portrait of the associated clock sequence. Some
sequences can yield visually appealing lengthy periodic behavior.
The phase portraits showing the ``beat'' (the clock) of $c_{16}$,
$c_{260}$, and $c_{138}$ appear in Figures
\ref{fig:beat16}-\ref{fig:beat138}.
\begin{figure}[th]
\begin{center}
\epsfxsize=100mm \epsfbox{beat16.eps}
\end{center}
\caption{The `beat' of $c_{16}$.}
\label{fig:beat16}
\end{figure}
\begin{figure}[th]
\begin{center}
\epsfxsize=100mm \epsfbox{beat260.eps}
\end{center}
\caption{The `beat' of $c_{260}$.}
\label{fig:beat260}
\end{figure}
\begin{figure}[th]
\begin{center}
\epsfxsize=100mm \epsfbox{beat138.eps}
\end{center}
\caption{The `beat' of $c_{138}$.}
\label{fig:beat138}
\end{figure}
Each of these phase portraits was drawn by computing the first ten
thousand terms of the sequence and the associated clock sequence.
Then, the next ten thousand terms of the associated clock sequence
were plotted as points $(t_k(n),t_k(n+1))$. If the clock has
become periodic by this point, revealing the underlying structure
of the sequence, the phase portrait then reveals a closed orbit
such as those in Figures \ref{fig:beat16}-\ref{fig:beat138}. On
the other hand, if the ``beat'' is still irregular, no apparent
order is discernable in the phase portrait. Once the underlying
structure is revealed, we may make conjectures concerning
equivalences with known sequences or
conjectures about the behavior of unknown sequences.
These conjectures are then often easily proved
(although when the period is long, the proof is tedious). As an
easy example we have
\begin{prop}
$c_4(n) = \lfloor
\frac{n+1}{2} \rfloor$.
\end{prop}
\begin{proof}
It is sufficient to prove that \beas c_4(n) = \lfloor
\frac{n+1}{2} \rfloor. \eeas We may easily verify that this is
true for $n=1$ to $n=6$. Now, assume it is true for $k = 1 \ldots
n$. Consider $c_4(n+1)$. We must show \beas c_4(n+1) = \lfloor
\frac{n+2}{2} \rfloor. \eeas But, \beas c_4(n+1) = c_4(4 c_4(n)
\bmod n) + c_4(n+1 - 4 c_4(n) \bmod n). \eeas By hypothesis \beas
c_4(n) = \lfloor \frac{n+1}{2} \rfloor, \eeas and hence \beas
c_4(n+1) = c_4( 4 \lfloor \frac{n+1}{2} \rfloor \bmod n) + c_4(n+1
- 4 \lfloor \frac{n+1}{2} \rfloor \bmod n). \eeas But, $4 \lfloor
\frac{n+1}{2} \rfloor \bmod n$ is $0$ if $n$ is even and $2$ if
$n$ is odd. So, \beas c_4(n+1) = c_4(1) + c_4(n) = 1+ \lfloor
\frac{n+1}{2} \rfloor, \eeas for $n$ even and \beas c_4(n+1) =
c_4(2) + c_4(n-1) = 1 + \lfloor \frac{n}{2} \rfloor, \eeas for $n$
odd. From which it follows directly that \beas c_4(n+1) = \lfloor
\frac{n+2}{2} \rfloor \eeas as desired.
\end{proof}
Note that we implicitly used the clock sequence in this proof. In fact, the result
can be restated as a result on the periodicity of $t_4$.

Randomly searching for $k$-sequences with the nice behavior of
$c_2$, $c_4$ or $c_{260}$, we develop the feeling that many
$k$-sequences are in fact irregular. To get a broader picture, we
compute a ``bifurcation diagram'' for the $c_k$. For each $k$, we
compute the first $5000$ terms, of both $c_k$ and $t_k$. Then, we
compute the density of $t_k$ in the interval $[0,2500]$. Finally,
we plot the negative log of this density versus $k$. Those
sequences with highly ordered clocks, and hence a clear underlying
structure, appear as peaks in this plot. Those with an irregular
``beat'' map roughly to zero. The bifurcation diagram for $k$
ranging from one to one-thousand appears in Figure
\ref{fig:bifurc}.
\begin{figure}
\begin{center}
\epsfxsize=100mm \epsfbox{bifurc.eps}
\end{center}
\caption{The bifurcation diagram for the $c_k$'s. Those $c_k$'s whose
order is apparent by examining $t_k$ appear as peaks.}
\label{fig:bifurc}
\end{figure}
Order appears to decrease with increasing $k$. Also the frequency of highly-ordered
sequences appears to decrease with increasing $k$.


\section{Open Questions and More Generalizations}
We have just scratched the surface of $k$-sequences. Many open
questions and further generalizations remain. One particularly
intriguing puzzle concerns irregular sequences such as $c_7$ and
$c_{13}$. Do the clocks of sequences such as $c_7$ or $c_{13}$
ever become periodic or do they always beat irregularly? Does
$c_k(n)/n$ tend to a limit for these sequences? Another question
concerns sequences with ``missing'' numbers such as $c_{11}$.
Computation reveals that $c_{11}$ misses $11$, $29$, $33$, $37$,
and $39$. Does $c_{11}$ miss infinitely many integers? What is the
sequence of integers missed? Yet another less precise question concerns order.
Is there a $k$-sequence that becomes irregular after a long period
of regularity? (The reader may wish to examine $c_{204}$ which
exhibits the opposite behavior.) We may also ask: What other known
sequences are lurking among the $k$'s? Finally, we note that
several authors have generalized the Conway-Hofstadter sequence in directions other than
the one presented here. The generalization presented here however
can be applied to those offered by Mallows \cite{Mallows}, Newman
\cite{Newman}, or Pinn \cite{Pinn}.
For example, Mallows \cite{Mallows}, introduces the sequence
\be
c(n)=c(c(n-2))+c(n-c(n-2)) \ee
as a generalization of the Conway-Hofstadter sequence. This generalization
bases the next term of the sequence on the second to last term of the sequence.
Multiplying the $c(n-2)$ terms by $k$ and computing modulo $n-1$, is a natural
generalization in the spirit of the $k$-sequences introduced here.
We hope the reader will be
intrigued by the preliminary results presented in this paper and
will be inspired to uncover new facts about the $c_k$'s or
generalize the work of Mallows, Newman, and Pinn, in the direction
suggested here.

\paragraph*{Acknowledgment}
Thanks to Julia Pelesko whose interest in Logo led to this work and to M.\
Tempel whose article \cite{Tempel} first introduced us to the Conway-Hofstadter sequence.
Thanks also to the anonymous referee for many useful comments and suggestions.

\begin{thebibliography}{20}

\bibitem{conwaytalk} J.H.\ Conway, Some Crazy Sequences, videotaped
talk at AT\&amp;T Bell Labs, July 15, 1988.

\bibitem{Mallows} C.L.\ Mallows, Conway's Challenge Sequence,
\textit{Amer. Math. Monthly} \textbf{98} (1991), 5--20.

\bibitem{NYT} M.W.\ Browne, Intellectual Duel: Brash Challenge,
Swift Response, \textit{The New York Times} Section C August 30 (1988), 1.

\bibitem{Kubo} T.\ Kubo and R.\ Vakil, On Conway's recursive
sequence, \textit{Disc. Math.} \textbf{152} (1996), 225--252.

\bibitem{Hofstadter} D.R.\ Hofstadter, Godel, Esher,Bach, Vintage
Books, New York, 1980, 137.

\bibitem{Newman} D.\ Newman, Problem E3274, \textit{Amer. Math.
Monthly} \textbf{95} (1988), 555.

\bibitem{Pinn} K.\ Pinn, A Chaotic Cousin of Conway's Recursive
Sequence, \textit{Exp. Math.} \textbf{9} (2000), 55--66.

\bibitem{Tempel} M.\ Tempel, Easy as $1 1 2 2 3$, \textit{Online publications
of the Logo Foundation}, el.media.mit.edu/logo-foundation/pubs/papers.

\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B37; Secondary 11B50.

\noindent \emph{Keywords:} Conway-Hofstadter, Fibonacci sequence,
nonlinear recursion.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A004001},
\seqnum{A004526},
\seqnum{A004396}, and
\seqnum{A037915}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received January 19 2004;
revised version received August 16 2004. 
Published in {\it Journal of Integer Sequences}, October 1 2004.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
\vskip .1in


\end{document}

                                                                                


\end{document}
</pre></body></html>