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\begin{document}
\title{Enumeration smooth Lego towers}
\maketitle
\thispagestyle{empty}

\section*{abstract}

%==============================================================
\section{Introduction}
In the following, we use the definitions from \cite{DZ}. Recall that ``Lego'' is a construction toy of interlocking plastic building blocks. For our purpose, we will consider them as rectangular tiles of dimensions $1\times a$ ($a \in \mathbb{N}$). A polyomino can be realized in terms of a Lego tower as follows: Assume that we have an infinite supply of Lego pieces of dimensions $1\times a$. Then every floor of the tower contains a finite (horizontal) sequence of pieces separated by gaps. A vertical sequence of floors constitutes a polyomino if the resulting configuration is connected. All the Lego towers depicted in Figure~\ref{fig1} are polyominoes.

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\caption{Examples of Lego towers.}\label{fig1}
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\begin{definition} A $k$-{\rm Lego tower} is a Lego tower with a finite number of floors, where each floor contains exactly one Lego piece of size $1\times a$ with $1\leq a\leq k$.
\end{definition}


For $k$-Lego towers, we introduce some more terminology. Given a $k$-Lego tower of two Lego pieces as described in Figure~\ref{fig2}, we call $\ell \in \mathbb{Z}$ the {\em left overhang} and $\ell' \in \mathbb{Z}$ the {\em right overhang} of the second floor with respect to the first floor; note that both overhangs can be positive or negative (in Figure~\ref{fig2}, $\ell <0$ and $\ell'>0 $) .
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\caption{Lego tower with two Lego pieces with $b-\ell' -\ell =a $.}\label{fig2}
\end{figure}

A Lego tower or $k$-lego tower is called {\em double side smooth} if the left overhang $\ell$ and the right overhang $\ell'$ satisfy $\ell,\ell'\in\{-1,1,0\}$, and is called {\em left side smooth} (respectively, {\em right side smooth}) if the left overhang $\ell$ (respectively, the right overhang $\ell'$) satisfy $\ell\in\{-1,1,0\}$ (respectively, $\ell'\in\{-1,1,0\}$).

\section{Enumeration of left side smooth Lego tower adn $k$-Lego tower}.

%-----------------------------
\begin{thebibliography}{50}

\bibitem{DZ}
D. Zeilberger, Automated counting of LEGO towers, {\it J. Differ. Equations Appl.} {\bf 5} (1999), 323--333.

\end{thebibliography}
\end{document} 