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\begin{center}{\maintitlesize \textbf{The \textsf{ThreeKPlusOne} Package\mbox{}}}\\
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\markright{\scriptsize \mbox{}\hfill The \textsf{ThreeKPlusOne} Package \hfill\mbox{}}
{\Huge Version 42\mbox{}}\\[1cm]
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{\Large \textbf{Dummy Auth{\"o}r  \mbox{}}}\\
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{\mbox{}\\
\small \noindent \textbf{Dummy Auth{\"o}r  }  Email: \href{mailto://3kplusone@dev.null} {\texttt{3kplusone@dev.null}}}\\
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{\small 
\section*{Copyright}
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{\copyright} 2000 The Author. 

 You can do with this package what you want.

 Really. \mbox{}}\\[1cm]
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\chapter{\textcolor{Chapter }{The $3k+1$ Problem}}\logpage{[ 1, 0, 0 ]}
\hyperdef{L}{X8769FB038733F6F4}{}
{
  
\section{\textcolor{Chapter }{Theory}}\label{sec:theory}
\logpage{[ 1, 1, 0 ]}
\hyperdef{L}{X8729B87B848E3F89}{}
{
  Let $k \in {\ensuremath{\mathbb N}}$ be a natural number. We consider the sequence $n(i, k), i \in {\ensuremath{\mathbb N}},$ with $n(1, k) = k$ and else $n(i+1, k) = n(i, k) / 2$ if $n(i, k)$ is even and $n(i+1, k) = 3 n(i, k) + 1$ if $n(i, k)$ is odd. 

 It is not known whether for any natural number $k \in {\ensuremath{\mathbb N}}$ there is an $m \in {\ensuremath{\mathbb N}}$ with $n(m, k) = 1$. 

 \textsf{ThreeKPlusOne} provides the function \texttt{ThreeKPlusOneSequence} (\ref{ThreeKPlusOneSequence}) to explore this for given $n$. If you really want to know something about this problem, see \cite{Wi98} or \href{http://www.ku.de/mgf/mathematik/lehrstuhlstatistik/team/dr-guenther-wirsching/} {\texttt{http://www.ku.de/mgf/mathematik/lehrstuhlstatistik/team/dr-guenther-wirsching/}} for more details (and forget this package). }

 
\section{\textcolor{Chapter }{Program}}\logpage{[ 1, 2, 0 ]}
\hyperdef{L}{X7A4605C07A94C9F6}{}
{
  In this section we describe the main function of this package. 

\subsection{\textcolor{Chapter }{ThreeKPlusOneSequence}}
\logpage{[ 1, 2, 1 ]}\nobreak
\hyperdef{L}{X7B1A31B8870EA092}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ThreeKPlusOneSequence({\mdseries\slshape k[, max]})\index{ThreeKPlusOneSequence@\texttt{ThreeKPlusOneSequence}}
\label{ThreeKPlusOneSequence}
}\hfill{\scriptsize (function)}}\\


 This function computes for a natural number \mbox{\texttt{\mdseries\slshape k}} the beginning of the sequence $n(i, k)$ defined in section \ref{sec:theory}. The sequence stops at the first $1$ or at $n(\mbox{\texttt{\mdseries\slshape max}}, k)$, if \mbox{\texttt{\mdseries\slshape max}} is given. 
\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@ThreeKPlusOneSequence(101);|
  "Sorry, not yet implemented. Wait for Version 84 of the package"
\end{Verbatim}
 }

 }

 }

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