\begin{itemize}
\item Algorithm~\footnote{
A.~J.~van Zanten and I.~N. Suparta.
\newblock Totally balanced and exponentially balanced gray codes.
 {\em Discrete Analysis and Operational Research}, 11:81--98, 2004.}:
inductive construction of $n$-bits Gray code given a $n-2$-bit Gray code
\item 
\emph{
Let $l$ be an even positive integer. Find 
$u_1, u_2, \dots , u_{l-2}, v$ (maybe empty) subsequences of $S_{n-2}$ 
such that $S_{n-2}$ is the concatenation of 
$
s_{i_1}, u_0, s_{i_2}, u_1, s_{i_3}, u_2, . . . , s_{i_l-1}, u_{l-2}, s_{i_l}, v
$
where $i_1 = 1$, $i_2 = 2$, and $u_0 = \emptyset$ (the empty sequence).
}
\item $\leadsto \#_n = \sum_{l'=1}^{2^{n-3}} {2^{n-2}-2 \choose 2l'-2}$ distinct 
  $u$ subsequences 
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
$n$ & 4&  5 & 6 & 7 & 8  \\  
\hline
$\#_n$ & 
1& 31 & 8191 & 5.3e8 & 2.3e18 \\
\hline
$\#'_n$ & 
1 & 15 & 3003 & 1.4e8 & 4.5e17  \\
\hline
\end{tabular}
\item A first simplification $\leadsto$ $\#'_n$
\end{itemize}

