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\begin{document}

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\begin{center}
\vskip 1cm
{\LARGE\bf A New Domino Tiling Sequence}
\vskip 1cm
\large Roberto Tauraso \\ 
Dipartimento di Matematica \\
Universit\`a di Roma ``Tor Vergata'' \\
via della Ricerca Scientifica\\
00133 Roma, Italy\\
{ \href{mailto:tauraso@mat.uniroma2.it}{\tt tauraso@mat.uniroma2.it} }
\vskip 1cm
\bf {Abstract}
\end{center}
{\em In this short note, we prove that the sequence \seqnum{A061646} 
from Neil Sloane's Online Encyclopedia of Integer Sequences is connected 
with the number of domino tilings of a holey square.}


\begin{section}{Introduction}
J. Bao discusses in \cite{Bao} the following conjecture due to Edward
Early (see also \cite{Pachter}): the number of tilings of a {\sl holey
square}, that is a $2n\times 2n$ square with a hole of size $2m\times
2m$ removed from the center, is $2^{n-m}\cdot k^2$ where $k$ is an odd
number which depends on $n$ and $m$. The conjecture has been proven for
$m=1,2,\dots, 6$ and numerical experiments indicate that it holds for
several other values of $m$. Here we are going to show that the
conjecture is verified also for $m=n-2$ and, perhaps what is more
surprising, that the corresponding odd numbers $k_n$ give a sequence
already contained in Sloane's Encyclopedia \cite{Sloane}, namely
\seqnum{A061646}.

Before proving this result we supply some definitions and notations.
The Fibonacci numbers \seqnum{A000045} are defined as the sequence of integers 
$$f_0 = 0, f_1 = 1, \mathrm{\ and\ } 
f_{n} = f_{n-1} +f_{n-2} \mathrm{\ for\ all\ } n\geq 2.$$
We will need Cassini's formula, one of the most famous Fibonacci identities: 
\begin{equation}
\label{cassini}
f_{n+1} f_{n-1}-f_{n}^2=(-1)^n\mathrm{\ for\ all\ } n\geq 1.
\end{equation}
The terms of
the sequence \seqnum{A061646} satisfy the recurrence relation given by 
$$a_0 = 1, a_1 = 1, a_2 = 3,\mathrm{\ and\ } 
a_{n} = 2 a_{n-1}+2 a_{n-2} - a_{n-3}  
\mathrm{\ for\ all\ } n\geq 3,$$
and the corresponding generating function is 
$$A(x)=\frac{1-x-x^2}{(1+x)(1-3x+x^2)}.$$
The number $a_n$ can be expressed in terms of the Fibonacci numbers, as
follows:
$$a_{n}=f_{n+1} f_{n-1}+f_{n}^2=2 f_{n}^2 + (-1)^n \mathrm{\ for\ all\ } n\geq 1.$$
The second formula is obtained from the first by using (\ref{cassini})  
and it says that $a_{n}$ is always an odd number.
The first $20$ values of the sequence $a_{n}$ compared
with the Fibonacci numbers are as follows:
\begin{center}
\begin{tabular}{ | r | r | r |}
\hline
$n$ &amp; $f_n$ &amp; $a_n$\\
\hline
0  &amp; 0 &amp; 1 \\
1  &amp; 1 &amp; 1 \\
2  &amp; 1 &amp; 3 \\
3  &amp; 2 &amp; 7 \\
4  &amp; 3 &amp; 19 \\
5  &amp; 5 &amp; 49 \\
6  &amp; 8 &amp; 129 \\
7  &amp; 13 &amp; 337 \\
8  &amp; 21 &amp; 883 \\
9  &amp; 34 &amp; 2311 \\
10 &amp; 55 &amp; 6051 \\
11 &amp; 89 &amp; 15841 \\
12 &amp; 144 &amp; 41473 \\
13 &amp; 233 &amp; 108577 \\
14 &amp; 377 &amp; 284259 \\
15 &amp; 610 &amp; 744199 \\
16 &amp; 987 &amp; 1948339 \\
17 &amp; 1597 &amp; 5100817 \\
18 &amp; 2584 &amp; 13354113 \\
19 &amp; 4181 &amp; 34961521 \\
\hline
\end{tabular}
\end{center}
\end{section}

\begin{section}{The Result}

\begin{Lemma}
For all $n,m\geq 1$ the number of domino tilings of the $L$-grid
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is 
\begin{equation}
\label{Lformula}
L_{n,m}=f_{n+1} f_{m}+f_{n} f_{m+1}.
\end{equation} 
Moreover, $L_{n,n-1}=a_n$ for all $n\geq 2$ and
\begin{equation}
\label{Lcassini}
L_{n,n-1}^2=L_{n,n}  L_{n-1,n-1} +1.
\end{equation}
\end{Lemma} 

\begin{proof} In order to prove (\ref{Lformula}), we start by covering the lower left corner. We can put there either a horizontal domino or a vertical one and then we can either cut along its smaller side or not.
Then we get the following four different configurations: 
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where the number of domino tilings of each configuration has been calculated
by recalling that the number of ways to tile a $2\times n$ rectangle is equal to $f_{n+1}$.

By (\ref{Lformula}), the identity (\ref{Lcassini}) is equivalent to
$$(f_{n+1} f_{n-1}+f_{n}^2)^2=(2 f_{n+1} f_{n})(2 f_{n-1} f_{n})+1=
4 f_{n+1} f_{n-1} f_{n}^2+1,$$
that is
$$(f_{n+1} f_{n-1}-f_{n}^2)^2=1$$
which holds by Cassini's formula (\ref{cassini}).
\end{proof}

Finally we show that, as expected by the previous conjecture, the number of domino tilings of the considered holey square is the product of $2^2$ by the square of an odd number. This odd number belongs to the sequence \seqnum{A061646}.


\begin{Theorem}
For all $n\geq 3$ the number of domino tilings of the holey square
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is
\begin{equation}
\label{Hformula} 
H_n=4 L_{n,n-1}^2=4 a_n^2.
\end{equation}
\end{Theorem} 

\begin{proof} By symmetry, the tilings which have a horizontal domino 
in the lower left corner are one half of the total number $H_n$:

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\noindent The first tiling can be continued by covering the upper right corner in 
these four ways:

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\noindent On the other hand the second tiling can be completed in only one way:

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\noindent Therefore
$$\frac{1}{2} H_{n}=L_{n,n-1}^2+L_{n,n} L_{n-1,n-1}+1$$
and by (\ref{Lcassini}) we obtain
$$\frac{1}{2} H_{n}=2 L_{n,n-1}^2=2 a_n^2$$
which gives us (\ref{Hformula}).
\end{proof}
\end{section}


\begin{thebibliography}{99}

\bibitem{Bao}
J. Bao, 
On the number of domino tilings of the rectangular grid, the holey square, and related problems, Final Report, Research Summer Institute at MIT, (1997).

\bibitem{Pachter}
L. Pachter, 
Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes, \emph{Electron. J. Comb.}, 
\textbf{4} (1997), 401--410.

\bibitem{Sloane} N. J. A. Sloane,
The On-Line Encyclopedia of Integer Sequences.
Published electronically at \htmladdnormallink{http://www.research.att.com/$\sim
$njas/sequences/}{http://www.research.att.com/~njas/sequences/}.
    
\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: \ \ Primary 05B45;
Secondary 11B37, 11B39.

\noindent \emph{Keywords:} domino tilings, Fibonacci numbers.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A000045} and \seqnum{A061646}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received March 26 2004;
revised version received  April 19 2004.
Published in {\it Journal of Integer Sequences}, April 27 2004.

\bigskip
\hrule
\bigskip

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


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