From sheubac@exchange.calstatela.edu Wed Feb 28 02:38:18 2007 Return-Path: Received: from mx1.calstatela.edu (mx1.calstatela.edu [130.182.1.41]) by graceland.math.uwaterloo.ca (8.13.8/8.13.8) with ESMTP id l1S7cB4A017804 for ; Wed, 28 Feb 2007 02:38:14 -0500 (EST) Received: from mx1.calstatela.edu (localhost.localdomain [127.0.0.1]) by localhost (Postfix) with SMTP id 933F32580B2; Tue, 27 Feb 2007 22:46:41 -0800 (PST) Received: from santafe8.academic.ad.calstatela.edu (santafe8.calstatela.edu [10.81.30.54]) by mx1.calstatela.edu (Postfix) with ESMTP id 4323E2580B3; Tue, 27 Feb 2007 22:46:40 -0800 (PST) Received: from owa2003.academic.ad.calstatela.edu ([10.81.30.52]) by santafe8.academic.ad.calstatela.edu with Microsoft SMTPSVC(6.0.3790.1830); Tue, 27 Feb 2007 22:46:40 -0800 Received: from [192.168.0.101] ([206.117.204.154]) by owa2003.academic.ad.calstatela.edu with Microsoft SMTPSVC(6.0.3790.1830); Tue, 27 Feb 2007 22:46:26 -0800 In-Reply-To: <200702201802.l1KI289s029248@graceland.math.uwaterloo.ca> References: <200702201802.l1KI289s029248@graceland.math.uwaterloo.ca> Mime-Version: 1.0 (Apple Message framework v624) Content-Type: multipart/mixed; boundary=Apple-Mail-26-1064340800 Message-Id: From: Silvia Heubach Subject: Re: referee report on your paper Date: Tue, 27 Feb 2007 22:44:28 -0800 To: Jeffrey Shallit X-Mailer: Apple Mail (2.624) X-OriginalArrivalTime: 28 Feb 2007 06:46:26.0477 (UTC) FILETIME=[2B1E95D0:01C75B04] X-Greylist: Delayed for 00:51:29 by milter-greylist-2.0 (graceland.math.uwaterloo.ca [129.97.75.152]); Wed, 28 Feb 2007 02:38:17 -0500 (EST) X-UUID: 8ac54261-2584-4306-b98f-2f636e6d0724 Status: RO X-Status: X-Keywords: X-UID: 5630 --Apple-Mail-26-1064340800 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; format=flowed Hi Jeffrey, I have sent this file and the eps files to a friend with a PC (as opposed to Mac), and it compiled on his machine, so I am hoping that it will do the same for you. Sorry for this inconvenience, this is the first time I had this problem. Hope it is fixed now. --Apple-Mail-26-1064340800 Content-Transfer-Encoding: quoted-printable Content-Type: application/x-tex; x-mac-hide-extension=yes; x-mac-type=54455854; x-unix-mode=0644; x-mac-creator=54655873; name="Ls-Squares-tilings 2-26-07-eps.tex" Content-Disposition: attachment; filename="Ls-Squares-tilings 2-26-07-eps.tex" \documentclass[12pt]{amsart}=0A\input{epsf}=0A= \usepackage{amssymb,latexsym,=20graphicx}=0A=0A=0A= \setlength{\unitlength}{1mm}=0A\topmargin=200=20pt=20\textheight=20= 46\baselineskip=20\advance\textheight=20by=0A\topskip=20= \setlength{\parindent}{0pt}=20\setlength{\parskip}{5pt=20plus=202pt=0A= minus=201pt}=20\setlength{\textwidth}{155mm}=0A= \setlength{\oddsidemargin}{5.6mm}=0A\setlength{\evensidemargin}{5.6mm}=0A= \numberwithin{equation}{section}=0A= \newtheorem{theorem}{Theorem}[section]=0A= \newtheorem{proposition}[theorem]{Proposition}=0A= \newtheorem{corollary}[theorem]{Corollary}=0A= \newtheorem{definition}[theorem]{Definition}=0A= \newtheorem{conjecture}[theorem]{Conjecture}=0A= \newtheorem{remark}[theorem]{Remark}=0A= \newtheorem{lemma}[theorem]{Lemma}=0A\newtheorem{fact}[theorem]{Fact}=0A= \newtheorem{example}[theorem]{Example}=0A= \newtheorem{axiom}[theorem]{Axiom}=0A= \newtheorem{property}[theorem]{Property}=0A= \newtheorem{problem}[theorem]{Problem}=0A= \newtheorem{notation}[theorem]{Notation}=0A\begin{document}=0A= \pagenumbering{arabic}=0A\pagestyle{headings}=0A= \def\sof{\hfill\rule{2mm}{2mm}}=0A\def\NN{\mathbb{N}}=0A= \def\mn{\mbox{-}}=0A\def\t{$\times$}=0A= \newcommand{\supsc}[1]{\ensuremath{^{\small=20\mbox{#1}}}}=0A=0A=0A= \title{Tiling=20with=20Ls=20and=20Squares}=0A\maketitle=0A=0A= \begin{center}=0APhyllis=20Chinn,=20Ralph=20Grimaldi=20and=20Silvia=20= Heubach=0A\end{center}=0A=0A\begin{center}=0A{Department=20of=20= Mathematics,=20Humboldt=20State=20University,=0AArcata,=20CA=2095521}=0A=0A= =20=20=20=20=20=20=20=20{\tt=20phyllis@humboldt.edu}\\[10pt]=0A=0A= {Department=20of=20Mathematics,=20Rose-Hulman=20Institute=20of=20= Technology\\=0ATerre=20Haute,=20IN=2047803}=0A=0A=20=20=20=20=20=20=20=20= {\tt=20ralph.grimaldi@rose-hulman.edu}\\[10pt]=0A=0A{Department=20of=20= Mathematics,=20California=20State=20University=20Los=20Angeles\\=0ALos=20= Angeles,=20CA=2090032}=0A=0A=20=20=20{\tt=20sheubac@calstatela.edu}=0A= \end{center}=0A=0A= %=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =0A\section*{Abstract}=0AWe=20consider=20tilings=20of=202$\times$$n$,=20= 3$\times$$n$,=20and=20$4\times=20n$=20boards=20with=201$\times$1=20= squares=20and=20L-shaped=20tiles=20covering=20an=20area=20of=20three=20= square=20units,=20which=20can=20be=20used=20in=20four=20different=20= orientations.=20For=20the=202$\times$$n$=20board,=20the=20recurrence=20= relation=20for=20the=20number=20of=20tilings=20is=20of=20order=20three=20= and,=20unlike=20most=20third=20order=20recurrence=20relations,=20can=20= be=20solved=20exactly.=20For=20the=203$\times$$n$=20and=20$4=20\times=20= n$=20board,=20we=20develop=20an=20algorithm=20that=20recursively=20= creates=20the=20basic=20blocks=20(tilings=20that=20cannot=20be=20split=20= vertically=20into=20smaller=20rectangular=20tilings)=20of=20size=20= 3$\times$$k$=20and=204$\times$$k$=20from=20which=20we=20obtain=20the=20= generating=20function=20for=20the=20total=20number=20of=20tilings.=20We=20= also=20count=20the=20number=20of=20L-shaped=20tiles=20and=201$\times$1=20= squares=20in=20all=20the=20tilings=20of=20the=202$\times$$n$=20and=20= 3$\times$$n$=20boards=20and=20determine=20which=20type=20of=20tile=20is=20= dominant=20in=20the=20long=20run.=0A=0A{\bf=20Keywords:=20}=20Tilings,=20= =20trominoes,=20recurrence=20relations,=20generating=20functions=0A=0A=0A= {\bf=20AMS=20Classification:}=2052C20,=2052C22=0A=0A=0A= %=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =0A\section{Introduction}=0ATiling=201$\times$$n$=20boards=20with=20= 1$\times$1=20squares=20(henceforth=20referred=20to=20simply=20as=20= squares)=20and=20dominoes=20(1$\times$2=20tiles)=20leads=20to=20a=20= connection=20with=20the=20Fibonacci=20numbers=20(see=20for=20= example~\cite{=20BenQui2003,=20BriCarChi1996}).=20These=20types=20of=20= tilings=20have=20also=20been=20of=20interest=20in=20chemistry=20and=20= statistical=20mechanics,=20where=20squares=20and=20dominoes=20are=20= referred=20to=20as=20monomers=20and=20dimers=20(see=20for=20= example~\cite{Caz1997,=20Kas1961,=20Ken1997,=20KenRanSin1996,=20=20=20= Rog1979,=20TemFis1961}).=20There=20are=20many=20possible=20extensions=20= to=20these=20most=20basic=20types=20of=20tilings=20(see=20for=20= example~\cite{BriChiHol1994,Har1994,=20HarChi1994,=20=20Heu1999,=20= HeuChi2001,=20HocRei2000}).=20=20Here,=20we=20consider=20tiling=20= $2$$\times$$n$,=20$3$$\times$$n$,=20and=20$4$$\times$$n$=20boards=20with=20= two=20types=20of=20tiles:=20=20squares=20and=20L-shaped=20tiles,=20or=20= Ls=20(covering=20an=20area=20of=20three=20square=20units),=20where=20the=20= Ls=20can=20be=20used=20in=20four=20different=20orientations.=20%Our=20= initial=20interest=20came=20from=20the=20fact=20that=20for=20the=20= 2$\times$$n$=20board,=20the=20recurrence=20for=20the=20number=20of=20= tilings=20leads=20to=20a=20recurrence=20of=20order=20three=20which=20can=20= be=20solved=20exactly.=0A=0AIn=20Section~\ref{basic}=20we=20present=20= notation=20and=20some=20general=20results=20on=20how=20the=20quantities=20= of=20interest=20are=20related=20to=20each=20other.=20Section~\ref{2n}=20= contains=20the=20results=20for=20the=20$2\times=20n$=20boards.=20We=20= obtain=20explicit=20results=20for=20the=20number=20of=20tilings,=20and=20= the=20number=20of=20Ls=20and=20squares=20in=20all=20the=20tilings=20of=20= a=20given=20size.=20We=20also=20derive=20a=20combinatorial=20expression=20= for=20the=20number=20of=20tilings=20in=20this=20case.=20In=20= Section~\ref{3n}=20we=20give=20=20results=20for=20the=20$3\times=20n$=20= boards,=20where=20the=20picture=20gets=20a=20lot=20more=20complex.=20We=20= develop=20an=20algorithm=20that=20creates=20recursively=20the=20{\it=20= basic=20blocks}=20(tilings=20that=20cannot=20be=20split=20vertically=20= into=20smaller=20rectangular=20tilings)=20of=20size=203$\times$$k$.=20= This=20allows=20us=20to=20count=20the=20number=20of=20basic=20blocks=20= of=20size=203$\times$$k$=20for=20all=20$k$,=20and=20in=20turn=20leads=20= us=20to=20the=20generating=20function=20for=20the=20total=20number=20of=20= tilings.=20Finally,=20in=20Section~\ref{mn},=20we=20obtain=20a=20= generating=20function=20for=20the=20number=20of=20tilings=20of=20the=20= $4\times=20n$=20boards,=20and=20relate=20our=20approach=20to=20grammars.=0A= =0A=0A=20= %=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =0A\section{Notation=20and=20basic=20results}\label{basic}=0A=0A=20We=20= will=20count=20the=20number=20of=20tilings,=20as=20well=20as=20the=20= number=20of=20squares=20and=20Ls=20in=20all=20tilings=20of=20an=20$m=20= \times=20n$=20board,=20=20where=20$m$=20denotes=20the=20vertical=20size=20= of=20the=20board=20(or=20number=20of=20rows),=20and=20the=20second=20= value=20(be=20it=20$n$,=20$k$,=20or=20$n-k$)=20refers=20to=20the=20= horizontal=20dimension=20(or=20number=20of=20columns).=20We=20will=20use=20= the=20following=20notation:=0A=0A\begin{center}=0A\begin{tabular}{lcl}=0A= =20=20$T_{m,n}$&=20=3D=20&=20number=20of=20tilings=20of=20size=20= $m$$\times$$n$=20with=20Ls=20and=20squares=20\\=0A=20$T_{m,n}^L=20$=20&=20= =3D=20&=20number=20of=20Ls=20in=20all=20tilings=20of=20size=20= $m$$\times$$n$=20\\=0A=20=20$T_{m,n}^S$=20&=20=3D=20&=20number=20of=20= squares=20in=20all=20tilings=20of=20size=20$m$$\times$$n$\\=0A=20= $B_{m,n}$=20&=20=3D=20&=20number=20of=20basic=20blocks=20of=20size=20= $m$$\times$$n$=20\\=0A=20$B_{m,n}^L$&=20=3D=20&=20number=20of=20Ls=20in=20= all=20basic=20blocks=20of=20size=20$m$$\times$$n$=20\\=0A=20=20= $B_{m,n}^S$&=20=3D=20&=20number=20of=20squares=20in=20all=20basic=20= blocks=20of=20size=20$m$$\times$$n$\\=0A=20\end{tabular}=0A=20= \end{center}=0A=0A=0AFor=20fixed=20$m$,=20we=20denote=20the=20generating=20= function=20$\sum_{n=3D0}^{\infty}a_{m,n}x^n$=20for=20a=20sequence=20= $\{a_{m,n}\}_{0}^{\infty}$=20by=20$G_{a_m}(x)$.=20Since=20we=20can=20= decompose=20any=20tiling=20of=20size=20$m=20\times=20n$=20into=20a=20= basic=20block=20of=20size=20$m=20\times=20k$=20on=20the=20left=20and=20a=20= smaller=20tiling=20of=20size=20$m=20\times=20(n-k)$=20following=20it,=20= we=20get=20this=20recursion:=0A\begin{eqnarray}=20\label{conv}=0AT_{m,n}=20= &=3D&=20\sum_{k=3D1}^n=20B_{m,k}\cdot=20T_{m,n-k}=20\quad=20\mbox{=20= for}=20\quad=20m,n=20\ge=201,=0A=20\end{eqnarray}=0A=20where=20we=20= define=20$T_{m,0}=3D1$=20for=20any=20$m=20\ge=201$.=20(A=20tiling=20of=20= size=20$m$\t$n$=20can=20consist=20of=20just=20a=20basic=20block=20of=20= size=20$m\times=20n$.)=20Since=20the=20recursion=20for=20$T_{m,n}$=20is=20= a=20convolution,=20the=20respective=20generating=20functions=20multiply=20= (see=20for=20example~\cite{Wil1994},=20Section=202.2,=20Rule=203).=20= Multiplying=20Eq.~(\ref{conv})=20by=20$x^n$,=20summing=20over=20$n=20\ge=20= 1$,=20and=20using=20the=20definition=20of=20the=20generating=20function,=20= we=20obtain=0A\begin{eqnarray}=20\label{convgf}=0A%=20\nonumber=20to=20= remove=20numbering=20(before=20each=20equation)=0A=20=20G_{T_{m}}(x)-1=20= =3D=20G_{B_{m}}(x)G_{T_{m}}(x)=20\Rightarrow=20=20= G_{T_{m}}(x)=3D\frac{1}{1-G_{B_{m}}(x)}.=0A=20\end{eqnarray}=0A=0AWe=20= will=20also=20count=20the=20number=20of=20squares=20and=20Ls=20in=20all=20= the=20tilings=20of=20an=20$m$\t$n$=20board.=20Looking=20at=20the=20total=20= area=20covered=20by=20all=20such=20tilings=20and=20splitting=20it=20up=20= according=20to=20areas=20covered=20by=20Ls=20and=20squares,=20we=20have=20= the=20following=20equation:=0A\begin{eqnarray}=20\label{area}=0A=20=20m=20= \cdot=20n=20\cdot=20T_{m,n}=20=3D=203=20T_{m,n}^L+=20T_{m,n}^S.=0A=20= \end{eqnarray}=0A=20Therefore,=20we=20have=20to=20count=20only=20one=20= of=20the=20two=20types=20of=20tiles.=20Again,=20we=20decompose=20each=20= tiling=20into=20a=20basic=20block=20and=20a=20smaller=20tiling.=20For=20= each=20such=20basic=20block,=20we=20get=20all=20the=20squares=20in=20the=20= tilings=20of=20the=20smaller=20size,=20and=20then,=20for=20each=20such=20= smaller=20tiling,=20we=20get=20the=20number=20of=20squares=20in=20the=20= basic=20block.=20Thus,=20for=20$m=20\ge=202$=20and=20$n=20\ge=201$,=0A= \begin{eqnarray}=20\label{Ss}=0A=20T_{m,n}^S=20=3D=20\sum_{k=3D1}^n=20= B_{m,k}=20\cdot=20T_{m,n-k}^S=20+=20\sum_{k=3D1}^n=20B_{m,k}^S=20\cdot=20= T_{m,n-k}.=0A=20=20\end{eqnarray}=0AOnce=20more,=20this=20is=20a=20= convolution,=20and=20we=20obtain=0A\begin{eqnarray}=20\label{Ssgf}=0A= \quad=20\quad=20= G_{T_m^S}(x)=3DG_{B_m}(x)G_{T_m^S}(x)+G_{B_m^S}(x)G_{T_m}(x)\;=20= \Rightarrow=20\;=20= G_{T_m^S}(x)=3D\frac{G_{B_m^S}(x)G_{T_m}(x)}{1-G_{B_m}(x)}.=0A=20=20= \end{eqnarray}=0AThe=20same=20type=20of=20argument=20gives=20these=20= results=20for=20Ls:=0A\begin{eqnarray}=20\label{Ls}=0A\quad=20\quad=20=20= T_{m,n}^L=20=3D=20\sum_{k=3D1}^n=20B_{m,k}=20\cdot=20T_{m,n-k}^L=20+=20= \sum_{k=3D1}^n=20B_{m,k}^L=20\cdot=20T_{m,n-k}=20\;=20\mbox{and}=20\;=20=20= G_{T_m^L}(x)=3D\frac{G_{B_m^L}(x)G_{T_m}(x)}{1-G_{B_m}(x)}.=0A=20=20= \end{eqnarray}=0A=0A%-------------New=20Section-------------------=0A=0A= \section{Tiling=202$\times$$n$=20boards}=20\label{2n}=0AWe=20first=20= look=20at=20the=20number=20of=20basic=20blocks=20of=20size=20= 2$\times$$k$.=20Clearly,=20$B_{2,1}=3D1$,=20$B_{2,2}=3D4$,=20and=20= $B_{2,3}=3D2$.=20There=20are=20no=20basic=20blocks=20for=20$k=20>=203$.=20= Figure~\ref{bb2}=20shows=20all=20the=20basic=20blocks=20of=20size=20= 2$\times$$k$.=0A=0A\begin{center}=0A\begin{figure}[h]=0A= \epsfxsize=3D400.0pt=20\epsffile{2bb.eps}=0A%\includegraphics{2bb}=0A= \caption{The=20basic=20blocks=20of=20size=202$\times$$k$}=20\label{bb2}=0A= \end{figure}=0A\end{center}=0A\vspace{-15pt}=0A=0AUsing=20= Eq.~(\ref{conv}),=20we=20get=20a=20recursion=20for=20$T_{2,n}$=20for=20= $n=20\ge=203$:=0A\begin{equation}\label{rec2}=0A= T_{2,n}=3DT_{2,n-1}+4T_{2,n-2}+2T_{2,n-3},=0A\end{equation}=0Awith=20= $T_{2,0}=3D1$,=20$T_{2,1}=3D1$,=20$T_{2,2}=3D5$=20(all=20the=20basic=20= blocks=20of=20size=20$2=20\times=202$=20and=20the=20tiling=20consisting=20= of=20four=20squares).=20The=20corresponding=20characteristic=20equation=20= is=20given=20by=20$x^3-x^2-4x-2=3D(x+1)(x^2-2x-2)=3D0$,=20and=20the=20= characteristic=20roots=20are=20-1,=20$1+=20\sqrt{3}$,=20and=20=20$1-=20= \sqrt{3}$.=20Consequently,=20= $T_{2,n}=3Dc_1(-1)^n+c_2(1+\sqrt{3})^n+c_3(1-\sqrt{3})^n$,=20and=20using=20= the=20initial=20conditions,=20we=20obtain=20$c_1=3D1$,=20$c_2=20=3D=20= 1/\sqrt{3}$,=20and=20$c_3=20=3D=20-1/\sqrt{3}$.=20Since=20the=20= resulting=20formula=20also=20gives=20the=20correct=20answers=20for=20= $n=3D1,2$=20and=20$3$,=20we=20have=20obtained=20an=20explicit=20formula=20= for=20$T_{2,n}$.=20The=20generating=20function=20for=20$T_{2,n}$=20can=20= easily=20be=20derived=20from=20the=20generating=20function=20for=20= $B_{2,n}$=20by=20using=20Eq.~(\ref{convgf}).=20=46rom=20= Figure~\ref{bb2},=20we=20can=20``read=20off''=20that=20= $G_{B_{2,n}}=3Dx+4x^{2}+3x^{3}$.=20We=20summarize=20these=20results=20in=20= the=20following=20theorem.=0A=0A\begin{theorem}=20\label{expl2n}=20The=20= number=20of=20tilings=20of=20size=202$\times$$n$=20with=20L-shaped=20= tiles=20and=20squares=20is=20given=20by=0A= $$T_{2,n}=3D(-1)^n+(1/\sqrt{3})((1+\sqrt{3})^n-(1-\sqrt{3})^n),=20\qquad=20= \mbox{=20for=20=20}=20n=20\ge=200,$$=0Awith=20generating=20function=20= $G_{T_{2,n}}(x)=3D1/(1-x-4x^2-2x^3)$.=0A\end{theorem}=0A=0ANote=20that=20= the=20recurrence=20in=20Eq.~(\ref{rec2})=20is=20of=20the=20form=20given=20= in=20Combinatorial=20Theorem=204=20in~\cite{BenQui2003}.=20Thus,=20= tilings=20of=202$\times$$n$=20boards=20with=20Ls=20and=20squares=20are=20= in=20one-to-one=20correspondence=20with=20tilings=20of=20a=20= 1$\times$$n$=20board=20with=20tiles=20of=20length=20at=20most=203,=20= where=20there=20is=20one=20tile=20of=20size=201$\times$1,=20four=20= different=20colored=20tiles=20of=20size=201$\times$2,=20and=20two=20= different=20colored=20tiles=20of=20size=201$\times$3,=20as=20shown=20in=20= Figure~\ref{coltiles}.=20Obviously,=20the=20idea=20of=20mapping=20basic=20= blocks=20of=20size=20$m=20\times=20k$=20to=20colored=20tiles=20of=20size=20= $1=20\times=20k$=20is=20applicable=20no=20matter=20what=20the=20shape=20= of=20the=20tiles=20is,=20and=20the=20number=20of=20colors=20used=20for=20= a=20tile=20of=20size=20$1=20\times=20k$=20=20is=20equal=20to=20the=20= number=20of=20basic=20blocks=20of=20size=20$m=20\times=20k$.=0A=0A= \begin{center}=0A\begin{figure}[h]=0A\epsfxsize=3D350.0pt=20= \epsffile{coltiles.eps}=0A%\includegraphics{coltiles}=0A= \caption{Correspondence=20with=20colored=20tilings=20of=201$\times$$k$=20= rectangles}=20\label{coltiles}=0A\end{figure}=0A\end{center}=0A= \vspace{-15pt}=0A=0AThe=20values=20for=20$T_{2,n}$=20for=20= $n=3D0,\ldots,20$=20are=20given=20by=20$\{1$,=20$=201$,=20$=205$,=20$=20= 11$,=20$=2033$,=20$=2087$,=20$=20241$,=20$=20655$,=20$=201793$,=20$=20= 4895$,=20$=2013377$,=20$=2036543$,=20$=2099841$,=20$=20272767$,=20= $745217$,=20$=202035967$,=20$=205562369$,=20$=2015196671$,=20$=20= 41518081$,=20$=20113429503$,=20$=20309895169\}$.=20The=20signed=20= version=20of=20this=20sequence=20appears=20in=20~\cite{Sloane}=20as=20= A077917.=20No=20combinatorial=20explanation=20for=20the=20signed=20= sequence=20is=20given.=0A=0AWe=20now=20count=20the=20number=20of=20Ls=20= and=20squares=20for=20tilings=20of=20the=20$2$\t$n$=20board.=20One=20way=20= to=20do=20this=20is=20to=20think=20of=20a=20tiling=20as=20a=20sequence=20= of=20basic=20blocks.=20Let=20\begin{center}=0A\begin{tabular}{lll}=0A=20=20= $s$=20&=3D=20&=20the=20number=20of=20basic=20blocks=20of=20size=20= $2$\t$1$=20(singles),=20\\=0A=20=20$d$=20&=3D=20&=20the=20number=20of=20= basic=20blocks=20of=20size=20$2$\t$2$=20(doubles),=20and=20=20\\=0A=20=20= $t$=20&=3D=20&=20the=20number=20of=20basic=20blocks=20of=20size=20= $2$\t$3$=20(triples).=20\\=0A=20\end{tabular}=0A=20\end{center}=0ANote=20= that=20$s+2d+3t=20=3D=20n$=20and=20that=20$s+d+t=3Dn-d-2t$=20gives=20the=20= number=20of=20basic=20blocks=20in=20the=20tiling.=0A=0A\begin{theorem}=20= The=20number=20of=20squares=20and=20L-shaped=20tiles=20for=20tilings=20= of=20the=20$2$\t$n$=20board=20are=20given=20in=20combinatorial=20form=20= by=0A\begin{eqnarray*}=0A=20=20=20=20T_{2,n}^S=20&=20=3D&\sum_{t=20=3D=20= 0}^{n/3}=20\sum_{d=3D0}^{(n-3t)/2}=20= \binom{n-d-2t}{t}\binom{n-d-3t}{d}(2s=20+=20d)4^d=202^t,=20\\=0A= T_{2,n}^L=20&=20=3D&\sum_{t=20=3D=200}^{n/3}=20\sum_{d=3D0}^{(n-3t)/2}=20= \binom{n-d-2t}{t}\binom{n-d-3t}{d}(d+2t)4^d=202^t,=0A\end{eqnarray*}=0A= or,=20explicitly,=20by=0A\begin{eqnarray}=20\label{numS}=0A=20= T_{2,n}^S&=3D&=20= (2n-12)(-1)^n+\frac{2}{3}((9-5\sqrt{3})(1+\sqrt{3})^n+(9+5\sqrt{3})(1-\sqr= t{3})^n)=20\\=0A=20=20\nonumber=20=20&&=20\qquad=20= +\frac{n}{\sqrt{3}}((\sqrt{3}-1)(1+\sqrt{3})^n=20+=20= (\sqrt{3}+1)(1-\sqrt{3})^n)=0A=20\end{eqnarray}=0A=20and=0A= \begin{eqnarray}=20\label{numL}=0A=20T_{2,n}^L&=3D&=20= 4(-1)^n-\frac{2}{9}((9-5\sqrt{3})(1+\sqrt{3})^n+(9+5\sqrt{3})(1-\sqrt{3})^= n)=20\\=0A=20=20=20\nonumber=20&&=20\qquad=20= -\frac{n}{3}((1-\sqrt{3})(1+\sqrt{3})^n=20+=20= (1+\sqrt{3})(1-\sqrt{3})^n).=0A=20\end{eqnarray}=0AThe=20respective=20= generating=20functions=20are=0A= $$G_{T_2^S}(x)=3D\frac{2x(1+2x)}{(1+x)^2(1-2x-2x^2)^2}\;=20\mbox{=20and=20= }=20\;=20G_{T_2^L}(x)=3D\frac{4x^2}{(1+x)(1-2x-2x^2)^2}.$$=0A= \end{theorem}=0A=0A\begin{proof}=20The=20combinatorial=20formula=20for=20= the=20respective=20number=20of=20tiles=20is=20derived=20=20using=20the=20= fact=20that=20each=20tiling=20of=20size=20$2$\t$n$=20is=20made=20up=20by=20= a=20succession=20of=20basic=20blocks.=20We=20count=20the=20number=20of=20= squares=20and=20Ls,=20respectively,=20according=20to=20the=20number=20of=20= singles,=20doubles=20and=20triples=20in=20a=20particular=20tiling,=20= then=20sum=20over=20all=20possible=20{\em=20configurations}=20$(s,d,t)$.=20= =20If=20a=20tiling=20has=20=20configuration=20$(s,d,t)$,=20then=20there=20= are=20$S(s,d,t)=3D2s+d$=20squares=20and=20$L(s,d,t)=3Dd+2t$=20Ls=20in=20= that=20tiling.=20=20To=20count=20the=20number=20of=20tilings=20with=20= configuration=20$(s,d,t)$,=20we=20first=20select=20the=20$t$=20positions=20= for=20the=20triples=20out=20of=20the=20possible=20$n-d-2t$=20positions.=20= Next=20we=20select=20the=20$d$=20positions=20for=20the=20doubles=20out=20= of=20the=20remaining=20$(n-d-2t)-t$=20positions.=20The=20remaining=20$s=20= =3D=20n-2d-3t$=20positions=20are=20filled=20with=20singles.=20Once=20the=20= positions=20have=20been=20selected,=20we=20can=20now=20fill=20each=20= ``triple=20position''=20with=20one=20of=20the=20two=20types=20of=20= triples,=20and=20each=20``double=20position''=20with=20one=20of=20the=20= four=20types=20of=20doubles.=20Thus=20there=20are=20$\binom{n-d-2t}{t}=20= \binom{n-d-3t}{d}=201^s=204^d=202^t$=20tilings=20with=20configuration=20= $(s,t,d)$,=20containing=0A$=20= \sum_{s+2d+3t=3Dn}\binom{n-d-2t}{t}\binom{n-d-3t}{d}=201^s=204^d=202^t=20= S(s,d,t)$=20=20squares=20and=20$=20\sum_{s+2d+3t=3Dn}\binom{n-d-2t}{t}=20= \binom{n-d-3t}{d}=201^s=204^d=202^t=20L(s,d,t)$=20Ls.=20The=20summation=20= can=20be=20written=20as=20a=20double=20sum=20by=20realizing=20that=20= there=20can=20be=20at=20most=20$n/3$=20triples=20and=20at=20most=20= $(n-3t)/2$=20doubles,=20together=20with=20exactly=20$n-2d-3t$=20singles.=20= Substituting=20gives=20the=20stated=20result.=0A=0AAlternatively,=20= since=20we=20have=20basic=20blocks=20only=20for=20$k=3D1,=202,=203$,=20= we=20actually=20get=20a=20third=20order=20recurrence=20from=20= Equations~(\ref{Ss})=20and~(\ref{Ls})=20for=20either=20the=20squares=20= or=20Ls.=20However,=20we=20now=20obtain=20a=20nonhomogeneous=20= recurrence=20equation:=0A\begin{eqnarray}=0A%=20\nonumber=20to=20remove=20= numbering=20(before=20each=20equation)=0A=20=20T_{2,n}^S=20=20&=3D&=20= (T_{2,n-1}^S+4\cdot=20T_{2,n-2}^S+2=20\cdot=20T_{2,n-3}^S)+(2\cdot=20= T_{2,n-1}+4\cdot=20T_{2,n-2})=20\\=0A=20T_{2,n}^L=20&=3D&=20= (T_{2,n-1}^L+4\cdot=20T_{2,n-2}^L+2=20\cdot=20T_{2,n-3}^L)+(4\cdot=20= T_{2,n-2}+4\cdot=20T_{2,n-3})=0A=20\end{eqnarray}=0A=20We=20solve=20the=20= recursion=20for=20the=20Ls.=20Using=20Theorem~\ref{expl2n}=20we=20find=20= that=20the=20nonhomogeneous=20part=20$a_n=3D4\cdot=20T_{2,n-2}+4\cdot=20= T_{2,n-3}$=20has=20the=20explicit=20form=20= $a_n=3D\frac{\sqrt{3}+2}{\sqrt{3}}(1+\sqrt{3})^n+\frac{\sqrt{3}-2}{\sqrt{3= }}(1-\sqrt{3})^n$.=20The=20homogeneous=20equation=20is=20identical=20to=20= that=20for=20$T_{2,n}$,=20thus=20we=20get=20the=20roots=20$-1,=20= 1+\sqrt{3},1-\sqrt{3}$,=20as=20before.=20This=20gives=20a=20general=20= solution=20=20(see=20for=20example~\cite{Gri2004})=20of=20the=20form=0A= $$T_{2,n}^L=3Dc_1(-1)^n+c_2(1+\sqrt{3})^n+c_3(1-\sqrt{3})^n+A\cdot=20= n(1+\sqrt{3})^n=20+=20B\cdot=20n=20(1-\sqrt{3})^n.$$=0AWe=20use=20= Mathematica=20to=20solve=20the=20system=20of=20five=20equations=20that=20= results=20from=20substituting=20the=20initial=20conditions=20= $T_{2,0}^L=3D0$,=20$T_{2,1}^L=3D0$,=20$T_{2,2}^L=3D4$,=20$T_{2,3}^L=3D12$,= =20$T_{2,5}^L=3D52$=20and=20obtain=0A\begin{eqnarray*}=0A=20T_{2,n}^L&=3D&= =20= 4(-1)^n-\frac{2}{9}((9-5\sqrt{3})(1+\sqrt{3})^n+(9+5\sqrt{3})(1-\sqrt{3})^= n)=20\\=0A=20=20=20&&=20\qquad=20-\frac{n}{3}((1-\sqrt{3})(1+\sqrt{3})^n=20= +=20(1+\sqrt{3})(1-\sqrt{3})^n).=0A=20\end{eqnarray*}=0AUsing=20= Eq.~(\ref{area})=20and=20the=20result=20for=20$T_{2,n}^L$=20yields=20the=20= result=20for=20$T_{2,n}^S$.=20The=20generating=20functions=20follow=20= easily=20from=20Eqs.~(\ref{Ssgf})=20and~(\ref{Ls}),=20where=20= $G_{B_2^S}(x)=3D2x+4x^2=3D2x(1+2x)$=20and=20= $G_{B_2^L}(x)=3D4x^2+4x^3=3D4x^2(1+x)$=20=20can=20be=20read=20off=20= directly=20from=20Figure~\ref{bb2}.=20For=20example,=20the=20basic=20= block=20of=20size=20$1\times1$=20contains=202=20squares,=20and=20the=20= basic=20blocks=20of=20size=20$2\times2$=20contain=204=20squares.=20= \end{proof}=0A=0A=0AThe=20numbers=20of=20squares=20and=20Ls=20in=20all=20= $2$\t$n$=20tilings=20for=20$n=3D1,\ldots,15$=20are=20given=20by=20$\{2$,=20= $=208$,=20$=2030$,=20$=20108$,=20$=20354$,=20$=201152$,=20$=203614$,=20$=20= 11204$,=20$=2034170$,=20$=20103176$,=20$=20308598$,=20$=20916236$,=20= $2702834$,=20$=207929872$,=20$=2023155182\}$=20and=20$\{0$,=20$=204$,=20= $=2012$,=20$=2052$,=20$=20172$,=20$=20580$,=20$=201852$,=20$=205828$,=20= $=2017980$,=20$=2054788$,=20$=20165116$,=20$=20493316$,=20$=201463036$,=20= $=204312068$,=20$=2012641276\}$,=20respectively.=20Neither=20of=20these=20= sequences=20occurred=20previously=20in=20~\cite{Sloane}.=0A=0AWe=20also=20= note=20that=20the=20sequence=20$a_n=3DT_{2,n+1}+T_{2,n}$=20which=20= occurs=20in=20the=20nonhomogeneous=20part=20of=20the=20equation=20for=20= $T_{2,n}^L$=20satisfies=20$a_n=3D2(a_{n-1}+a_{n-2})$,=20since=0A= \begin{eqnarray*}=0A=20=20=20a_n&=3D&=20= (T_{2,n}+4\,T_{2,n-1}+2\,T_{2,n-2})+=20T_{2,n}\\=0A=20=20=20&=3D&=20= (2\,T_{2,n}+2\,T_{2,n-1})+(2\,T_{2,n-1}+2\,T_{2,n-2})=3D=20= 2(a_{n-1}+a_{n-2}).=0A\end{eqnarray*}=0AThe=20sequence=20$\{a_n/2=20= \}_{n=3D0}^\infty$=20occurs=20in~\cite{Sloane}=20as=20A028859,=20and=20= counts=20the=20number=20of=20words=20on=20the=20alphabet=20\{0,1,2\}=20= without=20adjacent=20zeros.=20A=20direct=20bijection=20between=20these=20= words=20and=20the=20set=20of=20tilings=20of=20consecutive=20size=20is=20= not=20obvious.=0A=0ARichard=20Anstee=20posed=20the=20question=20which=20= of=20the=20two=20types=20of=20tiles=20becomes=20dominant=20in=20the=20= long=20run,=20where=20dominant=20can=20have=20two=20meanings=20-=20in=20= terms=20of=20actual=20number=20of=20tiles,=20or=20in=20terms=20of=20area=20= covered=20by=20the=20specific=20type=20of=20tile.=20=46rom=20= Eqs.~(\ref{numS})=20and~(\ref{numL}),=20we=20find=20that=0A= $$T_{{2,n}}^{S}\sim=20= \frac{n(\sqrt{3}-1)}{\sqrt{3}}(1+\sqrt{3})^{n}\quad\mbox{and}\quad=0A= T_{{2,n}}^{L}\sim=20\frac{n(\sqrt{3}-1)}{3}(1+\sqrt{3})^{n}.$$=0AThus,=0A= $$\frac{T_{{2,n}}^{S}}{T_{{2,n}}^{L}}\sim=20\sqrt{3}=20\quad=20= \mbox{and}=20\quad=20\frac{T_{{2,n}}^{S}}{3\cdot=20= T_{{2,n}}^{L}}\sim1/\sqrt{3},$$=20i.e.,=20there=20are=20more=20squares=20= than=20Ls=20by=20a=20factor=20of=20$\sqrt{3}$,=20but=20the=20area=20= covered=20by=20Ls=20is=20larger=20than=20the=20area=20covered=20by=20= squares=20by=20the=20same=20factor.=0A=0A=0A%-----------------------NEW=20= SECTION-----------------------=0A=0A\section{Tiling=203$\times$$n$=20= boards}=20\label{3n}=0A=0ATiling=20the=202$\times$$n$=20board=20was=20= straightforward=20because=20there=20were=20only=20basic=20blocks=20of=20= size=202$\times$$k$=20for=20$k=3D1,=202$=20and=20$3$.=20This=20resulted=20= in=20a=20linear=20recursion=20of=20order=20three=20that=20can=20be=20= solved=20explicitly.=20However,=20for=20the=20$3$$\times$$n$=20board=20= we=20get=20basic=20blocks=20of=20all=20sizes.=20As=20before,=20we=20get=20= one=20basic=20block=20of=20size=203$\times$1,=20which=20consists=20=20of=20= three=20squares.=20There=20are=20eight=20basic=20blocks=20of=20size=20= $3\times$=202,=20as=20we=20can=20place=20the=20Ls=20in=20exactly=20two=20= positions=20for=20each=20of=20the=20four=20orientations.=20In=20= addition,=20there=20are=20two=20basic=20blocks=20that=20contain=20two=20= Ls=20each.=20Figure~\ref{bb32}=20shows=20the=20ten=20basic=20blocks=20of=20= size=203$\times$2.=0A=0A\begin{center}=0A\begin{figure}[h]=0A= \epsfxsize=3D350.0pt=20\epsffile{3-2bb.eps}=0A= %\includegraphics[scale=3D1.2]{3-2bb}=0A\caption{The=20ten=20basic=20= blocks=20of=20size=203$\times$2}=20\label{bb32}=0A\end{figure}=0A= \end{center}=0A\vspace{-15pt}=0A=0AThe=203$\times$3=20basic=20blocks=20= have=20two=20Ls=20each,=20and=20it=20becomes=20clear=20that=20we=20need=20= to=20use=20a=20systematic=20way=20to=20create=20the=20basic=20blocks=20= to=20ensure=20that=20there=20are=20no=20duplicates=20and=20that=20we=20= actually=20have=20found=20all=20of=20the=20basic=20blocks.=20We=20will=20= create=20them=20recursively,=20by=20extending=20to=20the=20right,=20= i.e.,=20we=20create=20the=20basic=20blocks=20of=20size=20= 3$\times$$(k+1)$=20from=20those=20of=20size=203$\times=20k$=20for=20$k=20= \ge=202$.=20Since=20we=20do=20not=20want=20to=20destroy=20the=20basic=20= block=20property,=20we=20cannot=20replace=20an=20L=20in=20the=20last=20= column;=20we=20can=20only=20remove=20one=20or=20more=20squares=20and=20= replace=20it/them=20with=20an=20L=20in=20such=20a=20way=20that=20the=20L=20= fits=20into=20the=20additional=20column.=20We=20denote=20each=20= 1$\times$1=20area=20of=20the=20tiling=20as=20a=20{\it=20cell}=20to=20= distinguish=20it=20from=20the=20square=20(tile).=0A=0A=0A{\bf=20Basic=20= Block=20Creation=20(BBC)=20Algorithm:}=0A\begin{enumerate}=0A\item=20= Start=20with=20all=20basic=20blocks=20of=20size=20$m=20\times=202$,=20= and=20determine=20the=20possible=20compositions=20of=20the=20last=20= column=20in=20terms=20of=20cells=20covered=20by=20Ls=20and=20squares.=0A= \item=20For=20each=20different=20composition=20(up=20to=20horizontal=20= symmetry),=20=20assign=20a=20type.=0A\item=20For=20each=20type,=20= determine=20all=20possible=20ways=20to=20extend=20a=20basic=20block=20of=20= that=20type=20by=20placing=20at=20least=20one=20L=20over=20one=20or=20= two=20squares=20in=20the=20last=20column,=20and=20then=20filling=20the=20= remaining=20cells=20in=20the=20newly=20created=20column=20with=20= squares.=0A\end{enumerate}=0A=0ANote=20that=20(3)=20ensures=20that=20the=20= basic=20block=20property=20is=20maintained,=20so=20only=20basic=20blocks=20= are=20created.=20On=20the=20other=20hand,=20given=20a=20basic=20block=20= of=20size=20$m=20\times=20(k+1)$,=20it=20must=20have=20at=20least=20one=20= L-shaped=20tile=20that=20spans=20the=20$k$th=20and=20($k+1$)st=20= columns.=20Taking=20away=20any=20L-shaped=20tiles=20that=20span=20the=20= two=20rightmost=20columns,=20as=20well=20as=20all=20the=20square=20tiles=20= of=20the=20rightmost=20column,=20and=20then=20placing=20square=20tiles=20= into=20any=20empty=20cells=20in=20the=20$k$th=20column=20creates=20the=20= basic=20block=20of=20size=20$m=20\times=20k$=20from=20which=20the=20= basic=20block=20of=20size=20$m=20\times=20(k+1)$=20was=20created=20by=20= extension.=20Thus,=20by=20exhaustively=20listing=20all=20the=20possible=20= types=20and=20their=20extensions,=20we=20can=20ensure=20that=20all=20= basic=20blocks=20of=20size=20$m=20\times=20(k+1)$=20are=20created=20from=20= those=20of=20size=20$m=20\times=20k$.=0A=0AIn=20the=20case=20of=20= $3\times=20k$=20basic=20blocks,=20these=20are=20the=20possible=20types:=0A= \begin{itemize}=0A=20=20=20=20\item=20{\it=20Type=20I}:=20The=20last=20= column=20has=20=20two=20adjacent=20cells=20covered=20by=20a=20single=20= L.=20(Note=20that=20we=20cannot=20have=20two=20nonadjacent=20cells=20= covered=20by=20two=20different=20Ls,=20as=20these=20Ls=20=20would=20have=20= to=20overlap=20in=20the=20center=20of=20the=20next-to-last=20column).=0A=20= =20=20=20\item=20{\it=20Type=20II}:=20The=20last=20column=20has=20two=20= nonadjacent=20cells=20covered=20with=20squares=20(thus=20the=20center=20= cell=20is=20covered=20by=20an=20L).=0A=20=20=20=20\item=20{\it=20Type=20= III}:=20The=20last=20column=20has=20two=20adjacent=20cells=20covered=20= with=20squares=20(thus=20either=20the=20top=20or=20bottom=20=20cell=20is=20= covered=20by=20an=20L).=0A=20=20=20\end{itemize}=0AThe=20only=20other=20= composition=20for=20a=20basic=20block=20of=20size=20$3\times=202$=20is=20= to=20have=20all=20cells=20=20covered=20with=20Ls,=20but=20this=20type=20= cannot=20=20be=20extended=20and=20therefore,=20is=20not=20assigned=20a=20= type.=20In=20Figure~\ref{bb32},=20the=20first=20four=20blocks=20are=20of=20= Type=20I,=20the=206\supsc{th}=20and=207\supsc{th}=20are=20of=20Type=20= II,=20and=20the=205\supsc{th}=20and=208\supsc{th}=20are=20of=20Type=20= III.=20The=20last=20two=20blocks=20do=20not=20have=20a=20type.=0A=0AWe=20= =20now=20apply=20the=20BBC=20algorithm.=20=20For=20each=20basic=20block=20= of=20Type=20I,=20there=20is=20exactly=20one=20way=20to=20extend,=20as=20= shown=20in=20Figure~\ref{typeIext}=20for=20a=20basic=20block=20of=20size=20= 3$\times$2.=20(For=20easy=20reference,=20the=20cell=20where=20the=20= square=20is=20replaced=20by=20an=20L=20is=20marked=20with=20a=20dot.)=20= The=20resulting=20basic=20block=20is=20again=20of=20Type=20I,=20since=20= two=20adjacent=20cells=20in=20the=20last=20column=20are=20covered=20by=20= an=20L.=0A=0A\begin{center}=0A\begin{figure}[h]=0A\epsfxsize=3D150.0pt=20= \epsffile{Iext.eps}=0A%\includegraphics{Iext}=0A\caption{Extension=20for=20= Type=20I}=20\label{typeIext}=0A\end{figure}=0A\end{center}=0A= \vspace{-15pt}=0A=0AFor=20Type=20II=20basic=20blocks,=20the=20extension=20= can=20be=20done=20either=20at=20the=20top=20or=20the=20bottom=20cell,=20= i.e.,=20each=20Type=20II=20basic=20block=20of=20size=203$\times$$k$=20= creates=20two=20basic=20blocks=20of=20size=203$\times$$(k+1)$,=20and=20= both=20are=20of=20Type=20I=20(since=20two=20adjacent=20cells=20in=20the=20= last=20column=20are=20covered=20by=20an=20L.)=20Again,=20for=20each=20of=20= these=20extensions=20there=20is=20exactly=20one=20way,=20as=20indicated=20= in=20Figure~\ref{typeIIext},=20for=20one=20of=20the=20basic=20blocks=20= of=20size=203$\times$2.=0A=0A\begin{center}=0A\begin{figure}[h]=0A= \epsfxsize=3D400.0pt=20\epsffile{IIext.eps}=0A%\includegraphics{IIext}=0A= \caption{Extension=20for=20Type=20II}=20\label{typeIIext}=0A\end{figure}=0A= \end{center}=0A\vspace{-15pt}=0A=0AFinally,=20we=20look=20at=20the=20= extension=20for=20Type=20III=20blocks.=20There=20are=20several=20= possibilities.=20If=20we=20replace=20the=20square=20in=20the=20middle=20= ``row'',=20then=20there=20are=20two=20ways=20to=20place=20the=20L,=20= pointing=20either=20upward=20or=20downward.=20In=20either=20case,=20the=20= new=20block=20is=20of=20Type=20I.=20If=20we=20replace=20the=20square=20= at=20the=20top=20or=20bottom=20(depending=20on=20the=20particular=20= block),=20which=20can=20be=20done=20in=20exactly=20one=20way,=20a=20Type=20= I=20basic=20block=20is=20created.=20Finally,=20since=20there=20are=20two=20= adjacent=20cells=20covered=20by=20squares,=20we=20can=20place=20the=20L=20= so=20that=20it=20covers=20those=20two=20cells.=20This=20can=20be=20done=20= in=20two=20ways,=20one=20of=20which=20creates=20a=20Type=20II=20block,=20= whereas=20the=20other=20one=20creates=20a=20Type=20III=20block.=20=20=20= Figure~\ref{typeIIIext}=20shows=20the=20different=20possibilities.=0A=0A= \begin{center}=0A\begin{figure}[h]=0A\epsfxsize=3D450.0pt=20= \epsffile{IIIext.eps}=0A%\includegraphics{IIIext}=0A\caption{Extension=20= for=20Type=20III}=20\label{typeIIIext}=0A\end{figure}=0A\end{center}=0A= \vspace{-15pt}=0A=0AFigure~\ref{3-3bb}=20shows=20the=20basic=20blocks=20= of=20size=203$\times$3.=20Note=20that=20the=20first=20fourteen=20blocks=20= are=20of=20Type=20I,=20the=2015\supsc{th}=20and=2016\supsc{th}=20are=20= of=20Type=20II=20and=20the=20last=20two=20are=20of=20Type=20III.=0A=0A= \begin{center}=0A\begin{figure}[h]=0A\epsfxsize=3D450.0pt=20= \epsffile{3-3bb.eps}=0A%\includegraphics{3-3bb}=0A\caption{The=20basic=20= blocks=20of=20size=203$\times$3}=20=20\label{3-3bb}=0A\end{figure}=0A= \end{center}=0A\vspace{-15pt}=0A=0AThe=20BBC=20algorithm=20can=20be=20= visualized=20from=20Figures~\ref{3-3bb}=20and~\ref{3-4bb},=20which=20= show=20the=20basic=20blocks=20of=20size=203$\times$3=20and=203$\times$4,=20= respectively.=20The=20first=20fourteen=20blocks=20in=20= Figure~\ref{3-4bb}=20are=20the=20extensions=20of=20the=20fourteen=20Type=20= I=20blocks=20of=20size=203$\times$3.=20The=20next=20four=20blocks=20= result=20from=20the=20two=20extensions=20each=20of=20the=20two=20Type=20= II=20blocks=20(15\supsc{th}=20and=2016\supsc{th}=20in=20= Figure~\ref{3-3bb}),=20which=20are=20also=20of=20Type=20I.=20The=20= remaining=20ten=20blocks=20result=20from=20the=20extensions=20of=20the=20= two=20Type=20III=20blocks.=20The=20first=20six=20extensions=20are=20Type=20= I=20blocks,=20followed=20by=20two=20extensions=20of=20Type=20II,=20and=20= finally=20two=20extensions=20of=20Type=20III.=0A=0A\begin{center}=0A= \begin{figure}[h]=0A\epsfxsize=3D400.0pt=20\epsffile{3-4bb.eps}=0A= %\includegraphics{3-4bb}=0A\caption{The=20basic=20blocks=20of=20size=20= 3$\times$4}=20\label{3-4bb}=0A\end{figure}=0A\end{center}=0A= \vspace{-15pt}=0A=0AUsing=20the=20BBC=20algorithm,=20we=20obtain=20an=20= explicit=20formula=20for=20the=20number=20of=20basic=20blocks=20of=20= size=203$\times$$k$,=20and=20from=20Eq.~(\ref{conv}),=20a=20generating=20= function=20for=20the=20number=20of=20tilings=20of=20size=203$\times$$n$.=0A= =0A\begin{theorem}=20\label{BT3}=20The=20number=20of=20basic=20blocks=20= of=20size=203$\times$$k$=20is=20given=20by=20$$B_{3,1}=3D1,=20= B_{3,2}=3D10,=20\mbox{=20and=20}=20B_{3,k}=3D10k-12=20\quad=20\mbox{for=20= }=20k=20\ge=203.$$=20The=20generating=20functions=20for=20the=20number=20= of=20basic=20blocks=20and=20the=20number=20of=20tilings=20are=20given=20= by=20$$G_{B_3}(x)=3D\frac{x+8x^2-x^3+2x^4}{(1-x)^2}\quad=20\mbox{and}=20= \quad=20G_{T_3}(x)=3D\frac{(1-x)^2}{1-3x-7x^2+x^3-2x^4}.$$=0A= \end{theorem}=0A=0A\begin{proof}First=20of=20all=20we=20note=20that=20= the=20BBC=20algorithm=20does=20not=20create=20duplicates.=20The=20= extensions=20(if=20there=20are=20more=20than=20one)=20of=20a=20given=20= basic=20block=20of=20size=203$\times$$k$=20produce=20different=20basic=20= blocks=20of=20size=203$\times$$(k+1)$=20as=20the=20last=20columns=20= differ=20(see=20Figures~\ref{typeIext},=20\ref{typeIIext},=20and=20= \ref{typeIIIext}).=20Extensions=20from=20different=20basic=20blocks=20of=20= size=203$\times$$k$=20may=20produce=20the=20same=20last=20column,=20but=20= will=20differ=20somewhere=20in=20the=20first=20$k$=20columns.=20The=20= algorithm=20also=20produces=20all=20basic=20blocks=20of=20a=20given=20= size,=20=20as=20the=20three=20types=20are=20the=20only=20ones=20that=20= can=20occur=20for=20$m=3D3$,=20and=20the=20extensions=20take=20into=20= account=20all=20possibilities.=20Hence=20all=20blocks=20of=20size=20= 3$\times$$(k+1)$=20are=20created.=0A=0ALet=20$b_{k,t}$=20be=20the=20= number=20of=20basic=20blocks=20of=20size=203$\times$$k$=20of=20type=20= $t=3DI,II,III$,=20for=20$k=20\ge=202$.=20Then,=20for=20$k=20\ge=203$,=20= $B_{3,k}=3Db_{k,I}+b_{k,II}+b_{k,III}$.=20=46rom=20Figure~\ref{bb32}=20= we=20can=20read=20off=20the=20initial=20conditions=20=20$b_{2,I}=3D4$,=20= $b_{2,II}=3D2$,=20and=20$b_{2,III}=3D2$,=20and=20the=20BBC=20algorithm=20= yields=20the=20following=20recursions=20for=20$k\ge=203$:=20= $$b_{k+1,III}=3Db_{k,III}(=3D\ldots=3Db_{2,III}=3D2),=20\quad=20= b_{k+1,II}=3Db_{k,III}=3D2=20$$=0Aand=20$$b_{k+1,I}=3Db_{k,I}+=202=20= b_{k,II}+=203=20b_{k,III}=3Db_{k,I}+10,$$=20from=20which=20the=20formula=20= for=20=20$B_{3,k}$=20follows.=0A=0ATo=20obtain=20the=20generating=20= functions,=20we=20use=20the=20initial=20conditions=20$B_{3,1}=3D1$=20= (the=20basic=20block=20consisting=20of=20all=20squares)=20and=20= $B_{3,2}=3D10$=20(see=20Figure~\ref{bb32}).=20Thus,=20\begin{eqnarray*}=0A= =20=20G_{B_3}(x)=3D\sum_{k\ge1}B_{3,k}x^k=20&=3D&=20x+10x^2+=20= \left(\left(\sum_{k\ge1}(10k-12)x^k\right)=20-=20= \left(-2x+8x^2\right)\right)\\=0A=20=20=20&=3D&=20= 3x+2x^2+10\sum_{k\ge1}k\,x^k=20-12\sum_{k\ge1}x^k\\=0A=20=20&=3D&=20= 3x+2x^2+10\frac{x}{(1-x)^2}-12\frac{x}{1-x}=3D=20= \frac{x+8x^2-x^3+2x^4}{(1-x)^2}.=0A=20=20\end{eqnarray*}=0A=20The=20= result=20for=20$=20G_{T_3}(x)$=20follows=20from=20Eq.~(\ref{convgf})=20= and=20simplifying.=0A\end{proof}=0A=0AThe=20values=20for=20$T_{3,n}$=20= for=20$n=3D0,\ldots,15$=20are=20given=20by=20$\{1$,=20$1$,=20$11$,=20= $39$,=20$195$,=20$849$,=20$3895$,=20$17511$,=20$79339$,=20$358397$,=20= $1620843$,=20$7326991$,=20\=0A$33127155$,=20$149766353$,=20$677103839$,=20= $3061202815\}.$=20This=20sequence=20did=20not=20previously=20occur=20=20= in~\cite{Sloane}.=0A=0ANow=20we=20will=20count=20the=20number=20of=20Ls=20= and=20squares=20in=20the=20tilings=20of=20a=20$3$\t$n$=20board.=20=20In=20= light=20of=20Eqs.~(\ref{Ss})=20and~(\ref{Ls}),=20we=20see=20that=20we=20= have=20to=20obtain=20the=20generating=20function=20for=20either=20the=20= number=20of=20Ls=20or=20the=20number=20of=20squares=20in=20all=20basic=20= blocks.=20Figures~\ref{3-3bb}=20and~\ref{3-4bb}=20suggest=20that=20each=20= of=20the=20basic=20blocks=20of=20size=20$3$\t$k$=20(except=20those=20for=20= $k=20=3D=201,=202$)=20have=20exactly=0Athree=20squares.=20Thus,=20we=20= will=20count=20the=20squares=20first,=20and=20then=20use=20= Eq.~(\ref{area})=20to=20obtain=20the=20result=20for=20the=20Ls.=0A=0A= \begin{theorem}=20The=20generating=20functions=20for=20the=20numbers=20= of=20squares=20and=20L-shaped=20tiles=20in=20all=20tilings=20of=20the=20= $3$\t$n$=20board=20are=20given=20by=0A= $$G_{T_3^S}(x)=3D\frac{3x(1-x)^2(1+6x+3x^2)}{(1-3x-7x^2+x^3-2x^4)^2}\;=20= \mbox{=20and=20}=20\;=20= G_{T_3^L}(x)=3D\frac{4x^2(3-3x+3x^2-4x^3+x^4)}{(1-3x-7x^2+x^3-2x^4)^2}.$$=0A= \end{theorem}=0A=0A\begin{proof}=20Using=20the=20BBC=20algorithm,=20we=20= can=20show=20that=20each=20basic=20block=20of=20size=20$3$\t$k$=20for=20= $k=20\ge=203$=20contains=20exactly=20three=20squares.=20= Figure~\ref{3-3bb}=20shows=20that=20the=20induction=20hypothesis=20is=20= true=20for=20$k=3D3$.=20In=20the=20extension=20process=20we=20always=20= obtain=20as=20many=20squares=20in=20the=20additional=20column=20as=20get=20= covered=20in=20the=20prior=20last=20column,=20i.e.,=20the=20number=20of=20= squares=20remains=20the=20same.=20Thus,=20$B_{3,k}^S=3D3\cdot=20B_{3,k}$=20= for=20$k=20\ge=203$.=20Actually,=20this=20formula=20also=20holds=20for=20= $k=3D1$.=20For=20$k=3D2$,=20we=20get=20$B_{3,k}^S=3D24=3D3\,B_{3,k}-6$.=20= Therefore,=20$G_{B_3^S}(x)=3D3\cdot=20G_{B_3}(x)-6x^2=20=3D=20= 3x(1+6x+3x^2)/(1-x)^2$,=20where=20the=20last=20equality=20follows=20from=20= Theorem~\ref{BT3}.=20=20Using=20Eq.~(\ref{Ssgf}),=20we=20obtain=20the=20= desired=20result=20for=20$G_{T_3^S}(x)$.=20We=20note=20in=20passing=20= that=20$B_{3,k}^S=3D3\,(10k-12)=3D30\,(k-1)-6$=20for=20$k=20\ge=203$=20= (from=20Theorem~\ref{BT3}).=20This=20result=20also=20holds=20for=20= $k=3D2$.=20Turning=20to=20the=20number=20of=20Ls,=20Eq.~(\ref{area})=20= gives=20$T_{3,n}^L=3Dn\cdot=20T_{3,n}-T_{3,n}^S/3$=20for=20$n=20\ge=20= 0$.=20In=20terms=20of=20generating=20functions,=20this=20translates=20to=0A= $G_{T_3^L}(x)=3Dx=20\cdot=20G_{T_3}'(x)-G_{T_3^S}(x)/3$=20(see=20for=20= example~\cite{Wil1994},=20Section=202.2,=20Rule=202),=20which=20gives=20= the=20desired=20result=20after=20simplification.=0A\end{proof}=0A=0AThe=20= values=20for=20$n=20=3D=201,\ldots=2015$=20for=20the=20numbers=20of=20= squares=20and=20Ls=20in=20$3$\t$n$=20tilings=20are=20given=20by=20$\{3$,=20= $=2030$,=20$=20171$,=20$=201044$,=20$=205691$,=20$=2030678$,=20$=20= 159891$,=20$=20821100$,=20$=204151511$,=20$=2020764590$,=20$=20= 102880755$,=20$=0A505866804$,=20$=202471159019$,=20$=2012004723878$,=20$=20= 58037429739\}$=20and=20$\{=200$,=20$=2012$,=20$=2060$,=20$=20432$,=20$=20= 2348$,=20$=2013144$,=20$=2069280$,=20$=20361012$,=20$=201841736$,=20$=20= 9286900$,=20$=2046303316$,=20$=20\=0A228903592$,=20$=201123242916$,=20$=20= 5477879120$,=20$=2026572232312\}$,=20respectively.=20These=20sequences=20= did=20not=20previously=20occur=20=20in~\cite{Sloane}.=0A=0ATo=20answer=20= the=20question=20about=20which=20type=20of=20tile=20dominates=20in=20the=20= case=20of=20the=20$3\times=20n$=20tilings,=20we=20need=20to=20use=20= results=20from=20complex=20analysis=20about=20the=20asymptotic=20= behavior=20for=20meromorphic=20functions.=20=20Recall=20that=20if=20$f$=20= is=20a=20meromorphic=20function=20in=20a=20region=20$\mathcal{R}$=20and=20= $z_{0}$=20is=20a=20pole=20of=20order=20$r$,=20$1=20\le=20r=20<=20= \infty$,=20of=20$f$,=20then=20in=20some=20punctured=20disk=20centered=20= at=20$z_{0}$,=20$f$=20has=20an=20expansion=20of=20the=20form=0A$$=20= f(z)=3D\sum_{j=3D1}^{r}\frac{a_{-j}}{(z-z_{0})^{j}}+=20= \sum_{j=3D0}^{\infty}a_{j}(z-z_{0})^{j}=3DPP(f;z_{0})=20+=20= \sum_{j=3D0}^{\infty}a_{j}(z-z_{0})^{j},$$=20and=20the=20principal=20= part=20$PP(f;z_{0})$=20is=20given=20by=20(see=20\cite{Wil1994},=20= Eq.~(5.2.4))=0A\begin{equation}\label{Wilfeq}=0APP(f;z_{0})=3D\sum_{n=20= \ge=200}z^{n}\left\{=20= \sum_{j=3D1}^{r}\frac{(-1)^{j}a_{-j}}{z_{0}^{n+j}}\binom{n+j-1}{j-1}\right= \}.=0A\end{equation}=0AIn=20addition,=20for=20poles=20of=20order=202,=20= the=20coefficients=20$a_{-1}$=20and=20$a_{-2}$=20for=20a=20function=20= $f(z)=20=3D=20g(z)/h(z)$=20at=20the=20pole=20$z_{0}$=20are=20given=20by=20= (see=20for=20example~\cite{MarHof1987},=20Proposition=204.1.4=20and=20= proof=20thereof)=0A\begin{equation}=20\label{coeff}=0A= a_{-1}=3DRes\left(\frac{g}{h},z_{0}\right)=3D2\frac{g'(z_{0})}{h''(z_{0})}= -\frac{2}{3}\frac{g(z_{0})h'''(z_{0})}{[h''(z_{0})]^{2}}=20\quad=20= \mbox{and}=20\quad=20a_{-2}=3D\frac{2g(z_{0})}{h''(z_{0})}.=0A= \end{equation}=0A=0AWe=20are=20now=20ready=20to=20state=20the=20= asymptotic=20result.=0A=0A\begin{corollary}=20\label{3asym}=20The=20= asymptotic=20expressions=20for=20the=20number=20of=20squares=20and=20= L-shaped=20tiles=20in=20all=20tilings=20of=20the=20$3$\t$n$=20board=20= are=20given=20by=0A\begin{eqnarray*}=0AT_{3,n}^S&=20\,\sim=20&=20= (0.234746=20+=200.558529\,=20n)=20c_{1}^{n}=20+(0.0641066=20+=20= 0.329504\,n)c_{2}^{n},=20\\=0AT_{3,n}^L=20&\,=20\sim=20&=20(-0.0782485=20= +=200.268102\,=20n)=20c_{1}^{n}=20+(-0.0213689=20+=200.460065\,n)=20= c_{2}^{n},=0A\end{eqnarray*}=0Awhere=20$=20c_{1}=3D4.52104$,=20$c_{2}=20= =3D=20-1.738438$,=20and=20the=20error=20term=20is=20of=20order=0A= $O\left(\left(0.504457+\epsilon\right)^{n}\right)$.=0A\end{corollary}=0A=0A= \begin{proof}=20Both=20generating=20functions=20have=20the=20same=20= denominator,=20and=20thus=20the=20same=20poles.=20Using=20Mathematica,=20= we=20find=20that=20there=20are=20four=20different=20roots,=20two=20real=20= ones,=20$z_{0}=3D=200.221188$=20and=20$z_{1}=3D=20-0.575250$,=20and=20= two=20complex=20ones=20with=20modulus=20$R=20=3D1.98233$.=20(Note=20that=20= Mathematica=20gives=20exact=20roots=20that=20are=20used=20in=20the=20= subsequent=20calculations.=20We=20give=20numerical=20values=20to=20six=20= digits=20precision.)=20Applying=20Theorem=205.2.1~\cite{Wil1994}=20twice=20= to=20the=20poles=20at=20the=20real=20roots,=20we=20obtain=0A= \begin{eqnarray*}=0A= [z^{n}]f(z)&=3D&[z^{n}]\left(PP(f,z_{0})+PP(f,z_{1})\right)+O\left(\left(\= frac{1}{R}+\epsilon\right)^{n}\right)\\=0A= &=3D&\sum_{i=3D0,1}\left(\frac{-a_{i,-1}}{z_{i}}+\frac{(n+1)a_{i,-2}}{z_{i= }^{2}}\right)\left(\frac{1}{z_{i}}\right)^{n}+O\left(\left(\frac{1}{R}+\ep= silon\right)^{n}\right),=0A\end{eqnarray*}=0Awhere=20$a_{i,-j}$=20= denotes=20the=20coefficient=20$a_{-j}$=20for=20the=20pole=20$z_{i}$.=0A= Using=20Eqs.~(\ref{Wilfeq})=20and=20(\ref{coeff}),=20we=20find=20that=20= for=20$G_{T_3^S}(z)$,=20$a_{0,-1}=3D0.0716171$,=20$a_{0,-2}=3D0.0273256$,=20= $a_{1,-1}=3D=20-=200.152670$,=20and=20$a_{1,-2}=3D0.109037$.=20For=20= $G_{T_3^L}(z)$,=20the=20respective=20values=20are=20$0.0766088$,=20= $0.0131167$,=20$-0.276945$,=20and=20$0.152241$.=20These=20values=20were=20= obtained=20by=20using=20the=20exact=20roots=20computed=20in=20= Mathematica.=20Simplifying=20the=20expressions=20gives=20the=20desired=20= asymptotic=20results.=0A\end{proof}=0A=0AThe=20approximation=20is=20= quite=20good=20for=20both=20the=20number=20of=20squares=20and=20the=20= number=20of=20Ls.=20=20For=20small=20values=20of=20$n$,=20rounding=20the=20= approximate=20value=20to=20the=20nearest=20integer=20gives=20the=20= correct=20result,=20and=20for=20larger=20values,=20the=20approximation=20= provides=20at=20least=2010=20significant=20digits.=20Using=20= Corollary~\ref{3asym},=20we=20obtain=0A= $$\frac{T_{{3,n}}^{S}}{T_{{3,n}}^{L}}\sim=20= \frac{0.5585}{0.2681}=3D2.08318\quad=20\mbox{and}=20\quad=20= \frac{T_{{2,n}}^{S}}{3\cdot=20T_{{2,n}}^{L}}\sim0.694393,$$=0Aand=20can=20= answer=20the=20question=20on=20which=20type=20of=20tile=20dominates.=20= The=20number=20of=20squares=20is=20approximately=20twice=20the=20number=20= of=20L-shaped=20tiles,=20but=20the=20area=20covered=20by=20the=20= L-shaped=20tiles=20is=20about=2050\%=20more=20than=20the=20area=20= covered=20by=20the=20squares.=0A=0A=0A=0A%-----------------------NEW=20= SECTION-----------------------=0A=0A\section{Tiling=20$m=20\times$$n$=20= boards}=20\label{mn}=0A=0ATrying=20to=20obtain=20results=20for=20$m=20>=20= 3$=20becomes=20increasingly=20difficult,=20as=20illustrated=20below=20= for=20the=20case=20$m=20=3D=204$.=20The=20BBC=20algorithm=20can=20still=20= be=20used=20for=20$m=20=3D=203$,=20but=20its=20limitation=20for=20larger=20= values=20of=20$m$=20becomes=20obvious.=20Not=20only=20does=20the=20= number=20of=20basic=20blocks=20increase=20quite=20dramatically,=20but=20= the=20number=20of=20types=20also=20increases=20rapidly.=20For=20the=20= $4\times=202$=20board,=20there=20are=2032=20basic=20blocks=20instead=20= of=20the=2010=20basic=20blocks=20of=20size=20$3\times=202$.=20= Figure~\ref{bb4n}=20shows=20the=20basic=20blocks=20up=20to=20symmetry=20= (horizontal=20reflection),=20where=20all=20but=20the=20last=20four=20= blocks=20also=20occur=20in=20reflected=20form.=0A=0A\begin{center}=0A= \begin{figure}[h]=0A\epsfxsize=3D400.0pt=20\epsffile{bb42.eps}=0A= %\includegraphics{bb42}=0A\caption{Basic=20blocks=20of=20size=20= 4$\times$2}=20\label{bb42}=0A\end{figure}=0A\end{center}=0A= \vspace{-15pt}=0A=0A=20There=20are=20now=20nine=20different=20end=20= shapes,=20where=20what=20matters=20is=20=20the=20position=20of=20the=20= squares,=20as=20this=20is=20where=20the=20extension=20takes=20place.=20= Table~1=20shows=20the=20possible=20types=20(up=20to=20horizontal=20= reflection):=20Types=20I=20and=20II=20both=20have=20three=20squares,=20= Types=20III=20-=20VI=20have=20two=20squares,=20Types=20VII=20and=20VIII=20= have=20one=20square,=20and=20Type=20IX=20has=20no=20square.=0A=0A= \begin{table}[htdp]=20\label{types}=0A\begin{center}=0A\caption{The=20= nine=20types=20of=20end=20shapes=20for=204$\times=20k$=20basic=20blocks}=0A= \begin{tabular}{|c|c|c|c|}=0AI=20\phantom{xxx}=20II=20&\phantom{x}=20III=20= \phantom{xxx}=20IV=20\phantom{xxxx}=20V=20\phantom{xxxx}=20VI=20= \phantom{x}&=20VII=20\phantom{xx}=20=20VIII=20&=20IX\\=20\hline=0A= \epsfxsize=3D75.0pt=20\epsffile{3s.eps}=20&=20\epsfxsize=3D165.0pt=20= \epsffile{2s.eps}=20&=20\epsfxsize=3D75.0pt=20\epsffile{1s.eps}&=20= \epsfxsize=3D30.0pt=20\epsffile{0s.eps}=0A%\includegraphics{3s}=20&=20= \includegraphics{2s}=20&=20\includegraphics{1s}=20&=20= \includegraphics{0s}=0A\end{tabular}=0A\end{center}=0A\end{table}=0A=0A= Type=20IX=20does=20not=20extend,=20but=20gets=20created=20from=20some=20= of=20the=20other=20types,=20whereas=20Types=20V=20and=20VI=20exist=20= only=20for=20$n=3D2$.=20They=20can=20extend,=20but=20this=20type=20of=20= end=20shape=20never=20gets=20created=20(since=20the=20two=20columns=20= form=20a=20basic=20block=20by=20themselves=20and=20thus=20cannot=20be=20= a=20part=20of=20a=20larger=20basic=20block).=20Applying=20the=20BBC=20= Algorithm=20described=20in=20Section~\ref{3n}=20for=20each=20of=20the=20= types=20we=20obtain=20Table~2,=20which=20describes=20how=20many=20basic=20= blocks=20of=20each=20type=20are=20created=20when=20extending=20a=20basic=20= block=20of=20a=20certain=20type.=20For=20example,=20a=20Type=20III=20= basic=20block=20will=20create=20one=20basic=20block=20each=20of=20Type=20= I,=20II=20and=20IV,=20and=20two=20basic=20blocks=20of=20Type=20III.=0A=0A= \begin{table}[htdp]=0A=0A\begin{center}=0A\caption{Results=20of=20the=20= BBC=20algorithm}=0A\begin{tabular}{|c|ccccccccc|}=0A\phantom{x}=20&=20= \multicolumn{9}{c|}{Extension=20Type}\\=20\hline=0AType=20&=20I=20&=20II=20= &=20III=20&=20IV=20&=20V=20&=20VI=20&=20VII=0A=20&=20VIII=20&=20IX\\=20= \hline=0A=20I=20&=201=20&=203=20&=203=20&=202=20&=20-=20&=20-=20&=201=20= &=203=20&=202\\=0A=20=20II=20&=201=20&=201=20&=203=20&=201=20&=20-=20&=20= -=20&=201=20&=201=20&=202\\=0A=20=20=20=20III=20&=201=20&=201=20&=202=20= &=201=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-\\=0A=20=20=20=20IV=20&=20-=20= &=20-=20&=202=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20&=201\\=0A=20V=20= &=20-=20&=202=20&=202=20&=202=20&=20-=20&=20-=20&=20-=20&=20-=20&=201\\=0A= =20=20VI=20&=20-=20&=20-=20&=202=20&=201=20&=20-=20&=20-=20&=20-=20&=20-=20= &=201\\=0A=20=20=20=20VII=20&=20-=20&=20-=20&=201=20&=201=20&=20-=20&=20= -=20&=20-=20&=20-=20&=20-\\=0A=20=20=20=20=20=20VIII=20&=20-=20&=20-=20&=20= 1=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-\\=0A=20=20=20=20=20=20=20= =20=20IX=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20&=20-=20= &=20-\\=0A\end{tabular}=0A\end{center}=0A\label{4exten}=0A\end{table}%=0A= =0ALet=20$\tilde{b}_{k,t}$=20denote=20the=20number=20of=20basic=20blocks=20= of=20size=204$\times=20k$=20of=20type=20$t=20=3D=20I,=20II,=20\ldots=20= IX$,=20for=20$k=20\ge=202$.=20=46rom=20Figure~\ref{bb4n},=20we=20can=20= read=20off=20the=20values=20for=20$\tilde{b}_{2,t}$,=20and=20then,=20= using=20Table~2,=20we=20can=20compute=20the=20values=20for=20= $\tilde{b}_{3,t}$,=20which=20are=20given=20in=20Table~3.=0A=0A= \begin{table}[htdp]=0A=0A\begin{center}=0A\caption{Initial=20conditions=20= for=20$\tilde{b}_{n,t}$=20}=0A\begin{tabular}{c|ccccccccc|c}=0AType=20&=20= I=20&=20II=20&=20III=20&=20IV=20&=20V=20&=20VI=20&=20VII=0A=20&=20VIII=20= &=20IX=20&=20total\\=20\hline=0A=20$\tilde{b}_{2,t}$=20&=202=20&=204=20&=20= 4=20&=203=20&=201&=202=20&=204=20&=208=20&=204&=2032\\=0A=20= $\tilde{b}_{3,t}$=20&=2010=20&=2016=20&=2050=20&=2020=20&=200=20&=200=20= &=206=20&=2010=20&=2018&=20130\\=0A=0A\end{tabular}=0A\end{center}=0A= \label{inicond}=0A\end{table}%=0A=0A=0A=0AUsing=20Table~2=20we=20can=20= also=20read=20off=20a=20system=20of=20recurrence=20relations=20for=20= $\tilde{b}_{n,t}$=20=20for=20$t=20=3D=20I,=20\ldots=20,=20IV,=20VII,=20= VIII$,=20and=20$IX$.=20For=20example,=20Column=204=20of=20Table~2=20= gives=20this=20relation:=0A$$=20\tilde{b}_{n+1,IV}=20=3D=202=20= \tilde{b}_{n,I}=20+=20= \tilde{b}_{n,II}+\tilde{b}_{n,III}+2\tilde{b}_{n,VII},=20$$=0Asince=20= $\tilde{b}_{n,V}=3D\tilde{b}_{n,VI}=3D0$=20for=20$n=20>=202$.=20Adding=20= the=20equation=0A$$=20B_{4,n+1}=3D=20\tilde{b}_{n+1,I}=20+=20= \tilde{b}_{n+1,II}+\tilde{b}_{n+1,III}+\tilde{b}_{n+1,IV}+\tilde{b}_{n+1,V= II}+\tilde{b}_{n+1,VIII}+\tilde{b}_{n+1,IX},$$=20and=20using=20the=20= initial=20conditions=20for=20$n=3D3$,=20we=20can=20use=20the=20= Mathematica=20function=20GeneratingFunction=20from=20the=20package=20= DiscreteMath$^{\backprime}$RSolve=20to=20compute=20the=20generating=20= function=20for=20$B_{4,n}$=20for=20$n=20\ge=203$.=20Adjusting=20this=20= generating=20function=20=20by=20adding=20in=20the=20terms=20$B_{4,1}$=20= and=20$B_{4,2}$,=20we=20obtain=20the=20following=20theorem=20for=20the=20= $4\times=20n$=20tilings,=20where=20the=20generating=20function=20for=20= $T_{4,n}$=20follows=20from=20Eq.~(\ref{convgf}).=0A=0A\begin{theorem}=20= The=20generating=20function=20for=20the=20number=20basic=20blocks=20and=20= the=20number=20of=20tilings=20of=20the=20$4$\t$n$=20board=20with=20= L-shaped=20tiles=20and=20squares=20is=20given=20by=0A= $$G_{B_4}(x)=3D\frac{z(1=20+=2028z=20-=204z^2=20-=2080z^3=20-=2032z^4=20= +=2074z^5=20-=2010z^6=20-=204z^7=20-=204z^8)}{1=20-=204z=20-=206z^2=20-=20= 10z^3=20-=208z^4=20-=204z^5}$$=0A=20=20=20=20=20=20=20=20and=0A= $$G_{T_4}(x)=3D\frac{1=20-=204=20z=20-=206=20z^2=20-=2010=20=20z^3=20-=20= 8=20z^4=20-=204=20z^5}{1=20-=205z=20-=2034=20z^2=20-=206=20z^3=20+=2072=20= z^4=20+=2028=20z^5=20-=2074=20z^6=20+10=20z^7=20+=204=20z^8=20+=204=20= z^{9}}.$$=0A\end{theorem}=0A=0AThe=20values=20for=20$n=20=3D=200,\ldots,=20= 15$=20for=20the=20number=20of=20tilings=20of=20the=20$4=20\times=20n$=20= board=20are=20given=20by=20$\{1$,=20$1$,=20$=2033$,=20$=20195$,=20= $2023$,=20$=2016839$,=20$=20151817$,=20$=201328849$,=20$=2011758369$,=20= $=20103628653$,=20$=20914646205$,=20$=208068452381$,=20$=0A71189251649$,=20= $=20628067760289$,=20$=205541284098945$,=20$=2048888866203241\}$.=20= Results=20for=20the=20numbers=20of=20Ls=20and=20squares=20are=20much=20= more=20difficult=20to=20obtain,=20as=20the=20basic=20blocks=20no=20= longer=20have=20a=20constant=20number=20of=20squares,=20and=20one=20= would=20have=20to=20set=20up=20a=20system=20of=20recurrence=20relations=20= for=20the=20number=20of=20squares=20according=20to=20the=20type=20of=20= basic=20blocks,=20similar=20to=20that=20for=20the=20basic=20blocks=20= themselves.=0A=0A=0A=0AAn=20alternative=20approach=20is=20to=20express=20= tiling=20problems=20in=20terms=20of=20grammars,=20which=20allows=20for=20= automation=20of=20the=20process~\cite{=20MarMerRoc2000,=20MerSprVer2000,=20= MerSprVer2002}.=20Similar=20to=20the=20extension=20process=20described=20= in=20Section~\ref{3n},=20in=20the=20grammar=20approach=20a=20tiling=20is=20= built=20up=20by=20placing=20tiles=20onto=20the=20board=20in=20= lexicographic=20order,=20describing=20for=20each=20partial=20tiling=20= (which=20has=20a=20ragged=20right=20edge)=20which=20tiles=20can=20be=20= placed=20in=20the=20next=20step.=20This=20process=20produces=20a=20= system=20of=20recurrence=20relations,=20which=20can=20be=20solved=20= using=20a=20computer=20program.=20Note=20that=20the=20underlying=20ideas=20= are=20similar=20to=20the=20extension=20algorithm,=20but=20that=20there=20= are=20also=20differences.=20The=20states=20used=20in=20the=20grammar=20= approach=20are=20not=20complete=20tilings=20except=20for=20the=20initial=20= state,=20and=20one=20cannot=20separate=20out=20the=20basic=20blocks=20= from=20a=20composite=20tiling=20which=20is=20made=20up=20from=20more=20= than=20one=20basic=20block.=20Even=20with=20the=20grammar=20approach,=20= the=20tiling=20problem=20is=20solvable=20only=20for=20small=20to=20= midsize=20problems,=20as=20the=20complexity=20increases=20exponentially.=20= Focusing=20on=20basic=20blocks=20cuts=20down=20the=20size=20of=20the=20= problem,=20but=20so=20far=20there=20is=20no=20automatic=20algorithm=20= that=20creates=20basic=20blocks=20only=20and=20for=20which=20the=20= methodology=20used=20by=20Merlini=20and=20her=20co-authors=20can=20be=20= applied.=0A=0A\vspace{0.3in}=0A=0A\begin{center}=0A{\small=20= {ACKNOWLEDGEMENT}}=0A\end{center}=0A=0AWe=20would=20like=20to=20thank=20= Richard=20Anstee=20for=20raising=20the=20question=20on=20which=20type=20= of=20tile=20dominates=20in=20the=20long=20run,=20and=20the=20anonymous=20= referee=20for=20an=20extremely=20careful=20reading=20of=20the=20= manuscript.=0A=0A=0A=0A= %=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =0A=0A=0A=0A\begin{thebibliography}{99}=0A=0A\bibitem{BenQui2003}=0A= A.~T.=20Benjamin=20and=20J.~J.=20Quinn.=0A\newblock=20{\em=20Proofs=20= that=20really=20count},=20volume~27=20of=20{\em=20The=20Dolciani=0A=20=20= Mathematical=20Expositions}.=0A\newblock=20Mathematical=20Association=20= of=20America,=20Washington,=20DC,=202003.=0A%\newblock=20The=20art=20of=20= combinatorial=20proof.=0A=0A\bibitem{BriCarChi1996}=0AR.~C.=20Brigham,=20= R.~M.=20Caron,=20P.~Z.=20Chinn,=20and=20R.~P.=20Grimaldi.=0A\newblock=20= A=20tiling=20scheme=20for=20the=20{Fibonacci}=20numbers.=0A\newblock=20= {\em=20J.=20Recreational=20Mathematics},=20{\bf=2028}=20(1996)=2010--17.=0A= =0A\bibitem{BriChiHol1994}=0AR.~C.=20Brigham,=20P.~Z.=20Chinn,=20L.~M.=20= Holt,=20and=20R.~S.=20Wilson.=0A\newblock=20Finding=20the=20recurrence=20= relation=20for=20tiling=20{$2\times=20n$}=20rectangles.=0A\newblock=20= {\em=20Congr.=20Numer.},=20{\bf=20105}=20(1994)=20134--138.=0A=0A= \bibitem{Caz1997}=0AF.~Cazals.=0A\newblock=20Monomer-dimer=20tilings.=0A= \newblock=20http://citeseer.ist.psu.edu/cazals97monomerdimer.html,=20= 1997.=0A=0A\bibitem{Gri2004}=0AR.~Grimaldi.=0A\newblock=20{\em=20= Discrete=20and=20Combinatorial=20Mathematics,=205th=20Edition}.=0A= \newblock=20Pearson=20Education,=20Inc.,=202004.=0A=0A\bibitem{Har1994}=0A= E.~O.=20Hare.=0A\newblock=20Tiling=20a=20{$3\times=20n$}=20area=20with=20= {C}uisenaire=20rods=20of=20length=20less=0A=20=20than=20or=20equal=20to=20= {$k$}.=0A\newblock=20{\em=20Congr.=20Numer.},=20{\bf=20105}=20=20(1994)=20= 33--45.=0A=0A\bibitem{HarChi1994}=0AE.~O.=20Hare=20and=20P.~Z.=20Chinn.=0A= \newblock=20Tiling=20with=20{C}uisenaire=20rods.=0A\newblock=20In=20{\em=20= Applications=20of=20Fibonacci=20numbers,=20Vol.\=206=20(Pullman,=20WA,=0A= =20=201994)},=20165--171.=20Kluwer=20Acad.=20Publ.,=20Dordrecht,=201996.=0A= =0A\bibitem{Heu1999}=0AS.~Heubach.=0A\newblock=20Tiling=20an=20= $m$-by-$n$=20area=20with=20squares=20of=20size=20up=20to=20$k$-by-$k$=20= with=0A=20=20$m=20\le=205$.=0A\newblock=20{\em=20Congr.=20Numer.},=20= {\bf=20140}=20=20(1999)=2043--64.=0A=0A\bibitem{HeuChi2001}=0AS.~Heubach=20= and=20P.=20Z.~Chinn.=0A\newblock=20Patterns=20arising=20from=20tiling=20= rectangles=20with=201-by-1=20and=202-by-2=0A=20=20squares.=0A\newblock=20= {\em=20Congr.=20Numer.},=20=20{\bf=20150}=20(2001)=20173--192.=0A=0A=0A=0A= \bibitem{HocRei2000}=0AR.=20Hochberg=20and=20M.=20Reid.=0A\newblock=20= 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\bibitem{MarMerRoc2000}=0AL.=20Martinelli,=20D.=20Merlini,=20et.al.=0A= \newblock=20A=20software=20to=20solve=20strip=20tiling=20problems.=0A= \newblock=20{\em=20Formal=20Power=20Series=20and=20Algebraic=20= Combinatorics=20(FPSAC,=20Moscow=202000)},=20801--805,=20Springer=20= Berlin,=202000.=0A=0A=0A\bibitem{MerSprVer2000}=0AD.=20Merlini,=20R.=20= Sprugnoli,=20and=20M.~C.=20Verri.=0A\newblock=20Strip=20tiling=20and=20= regular=20grammars.=0A\newblock=20{\em=20Theoretical=20Computer=20= Science},=20{\bf=20242}=20=20(2000)=20109--124.=0A=0A= \bibitem{MerSprVer2002}=0AD.=20Merlini,=20R.=20Sprugnoli,=20and=20M.~C.=20= Verri.=0A\newblock=20A=20strip-like=20tiling=20algorithm.=0A\newblock=20= {\em=20Theoretical=20Computer=20Science},=20{\bf=20282}=20=20(2002)=20= 337--352.=0A=0A\bibitem{Rog1979}=0AD.~G.=20Rogers.=0A\newblock=20An=20= application=20of=20renewal=20sequences=20to=20the=20dimer=20problem.=0A= \newblock=20In=20{\em=20Combinatorial=20mathematics,=20VI=20(Proc.=20= Sixth=20Austral.=20Conf.,=0A=20=20Univ.=20New=20England,=20Armidale,=20= 1978)},=20volume=20748=20of=20{\em=20Lecture=20Notes=20in=0A=20=20= Math.},=20143--153.=20Springer,=20Berlin,=201979.=0A=0A\bibitem{Sloane}=0A= N.=20J.=20A.=20Sloane.=0A\newblock=20The=20on-line=20encyclopedia=20of=20= integer=20sequences.=0A\newblock=20published=20electronically=20at=0A=20=20= http://www.research.att.com/$\sim$njas/sequences/,=202007.=0A=0A= \bibitem{TemFis1961}=0AH.=20N.=20V.=20Temperley=20and=20M.=20E.=20= Fisher.=0A\newblock=20Dimer=20problem=20in=20statistical=20mechanics,=20= an=20exact=20result.=0A\newblock=20{\em=20Philosophical=20Magazine},=20= {\bf=206}=20=20(1961)=201061--1063.=0A=0A\bibitem{Wil1994}=0AH.~S.=20= Wilf.=0A\newblock=20{\em=20Generatingfunctionology,=202nd=20Edition}.=0A= \newblock=20Academic=20Press,=20Inc.,=20=201994.=0A=0A= \end{thebibliography}=0A=20\vspace{0.2in}=0A=0ASequences=20mentioned=20= in=20this=20article:=20A077917=20and=20A028859.=0A=0A\end{document}=0A= 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def /Italic /Courier-Oblique findfont def /MathPictureStart { /Mimatrix matrix currentmatrix def gsave newpath Mleft Mbottom translate /Mtmatrix matrix currentmatrix def Plain Mfontsize scalefont setfont 0 setgray 0 setlinewidth } bind def /MathPictureEnd { grestore } bind def /MathSubStart { Momatrix Mgmatrix Mtmatrix Mleft Mbottom Mwidth Mheight 9 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 11 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def /Mleft 0 def /Mbottom 0 def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } bind def /MathSubEnd { /Mheight exch def /Mwidth exch def /Mbottom exch def /Mleft exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto lineto lineto fill } bind def /Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke } bind def /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash } bind def /Mfstroke { stroke currentdash pop 0 setdash } bind def /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def /Mvboxa { Mfixwid { Mvboxa1 } { dup Mwidthcal 0 exch { add } forall exch Mvboxa1 4 index 7 -1 roll add 4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def /Mbbox { 0 0 moveto false charpath flattenpath pathbbox newpath } bind def /Mmin { 2 copy gt { exch } if pop } bind def /Mmax { 2 copy lt { exch } if pop } bind def /Mrotshowa { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix Mshowa /Mtmatrix exch def /Mrot 0 def } bind def /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get show } for } bind def /Mshoway { 3 index Mwidthcal 5 1 roll 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get [ 6 index aload length 2 add -1 roll { pop Strform stringwidth pop neg exch add 0 rmoveto } exch kshow cleartomark } for pop } bind def /Mwidthcal { [ exch { Mwidthcal1 } forall ] [ exch dup Maxlen -1 add 0 1 3 -1 roll { [ exch 2 index { 1 index Mget exch } forall pop Maxget exch } for pop ] Mreva } bind def /Mreva { [ exch aload length -1 1 {1 roll} for ] } bind def /Mget { 1 index length -1 add 1 index ge { get } { pop pop 0 } ifelse } bind def /Maxlen { [ exch { length } forall Maxget } bind def /Maxget { counttomark -1 add 1 1 3 -1 roll { pop Mmax } for exch pop } bind def /Mwidthcal1 { [ exch { Strform stringwidth pop } forall ] } bind def /Strform { /tem (x) def tem 0 3 -1 roll put tem } bind def /Mshowa1 { 2 copy add 4 1 roll sub mul sub -2 div } bind def /MathScale { Mwidth Mheight Mlp translate scale /yscale exch def /ybias exch def /xscale exch def /xbias exch def /Momatrix xscale yscale matrix scale xbias ybias matrix translate matrix concatmatrix def /Mgmatrix matrix currentmatrix def } bind def /Mlp { 3 copy Mlpfirst { Mnodistort { Mmin dup } if 4 index 2 index 2 index Mlprun 11 index 11 -1 roll 10 -4 roll Mlp1 8 index 9 -5 roll Mlp1 4 -1 roll and { exit } if 3 -1 roll pop pop } loop exch 3 1 roll 7 -3 roll pop pop pop } bind def /Mlpfirst { 3 -1 roll dup length 2 copy -2 add get aload pop pop pop 4 -2 roll -1 add get aload pop pop pop 6 -1 roll 3 -1 roll 5 -1 roll sub dup /MsaveAx exch def div 4 1 roll exch sub dup /MsaveAy exch def div } bind def /Mlprun { 2 copy 4 index 0 get dup 4 1 roll Mlprun1 3 copy 8 -2 roll 9 -1 roll { 3 copy Mlprun1 3 copy 11 -3 roll /gt Mlpminmax 8 3 roll 11 -3 roll /lt Mlpminmax 8 3 roll } forall pop pop pop pop 3 1 roll pop pop aload pop 5 -1 roll aload pop exch 6 -1 roll Mlprun2 8 2 roll 4 -1 roll Mlprun2 6 2 roll 3 -1 roll Mlprun2 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setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto lineto lineto fill } bind def /Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke } bind def /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash } bind def /Mfstroke { stroke currentdash pop 0 setdash } bind def /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def /Mvboxa { Mfixwid { Mvboxa1 } { dup Mwidthcal 0 exch { add } forall exch Mvboxa1 4 index 7 -1 roll add 4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def /Mbbox { 0 0 moveto false charpath flattenpath pathbbox newpath } bind def /Mmin { 2 copy gt { exch } if pop } bind def /Mmax { 2 copy lt { exch } if pop } bind def /Mrotshowa { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix Mshowa /Mtmatrix exch def /Mrot 0 def } bind def /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get show } for } bind def /Mshoway { 3 index Mwidthcal 5 1 roll 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get [ 6 index aload length 2 add -1 roll { pop Strform stringwidth pop neg exch add 0 rmoveto } exch kshow cleartomark } for pop } bind def /Mwidthcal { [ exch { Mwidthcal1 } forall ] [ exch dup Maxlen -1 add 0 1 3 -1 roll { [ exch 2 index { 1 index Mget exch } forall pop Maxget exch } for pop ] Mreva } bind def /Mreva { [ exch aload length -1 1 {1 roll} for ] } bind def /Mget { 1 index length -1 add 1 index ge { get } { pop pop 0 } ifelse } bind def /Maxlen { [ exch { length } forall Maxget } bind def /Maxget { counttomark -1 add 1 1 3 -1 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bind def /MathPictureEnd { grestore } bind def /MathSubStart { Momatrix Mgmatrix Mtmatrix Mleft Mbottom Mwidth Mheight 9 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 11 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def /Mleft 0 def /Mbottom 0 def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } bind def /MathSubEnd { /Mheight exch def /Mwidth exch def /Mbottom exch def /Mleft exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto lineto lineto fill } bind def /Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke } bind def /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash } bind def /Mfstroke { stroke currentdash pop 0 setdash } bind def /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def /Mvboxa { Mfixwid { Mvboxa1 } { dup Mwidthcal 0 exch { add } forall exch Mvboxa1 4 index 7 -1 roll add 4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if 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index exec { 7 -3 roll 4 -1 roll } if 1 index 5 index 3 -1 roll exec { 4 1 roll pop 5 -1 roll aload pop pop 4 -1 roll aload pop [ 8 -2 roll pop 5 -2 roll pop 6 -2 roll pop 5 -1 roll ] 4 1 roll pop } { pop pop pop } ifelse } bind def /Mlp1 { 5 index 3 index sub 5 index 2 index mul 1 index le 1 index 0 le or dup not { 1 index 3 index div .99999 mul 8 -1 roll pop 7 1 roll } if 8 -1 roll 2 div 7 -2 roll pop sub 5 index 6 -3 roll pop pop mul sub exch } bind def /intop 0 def /inrht 0 def /inflag 0 def /outflag 0 def /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def /Minner { outflag 1 eq { /outflag 0 def /intop 0 def /inrht 0 def } if 5 index gsave Mtmatrix setmatrix Mvboxa pop grestore 3 -1 roll pop dup intop gt { /intop exch def } { pop } ifelse dup inrht gt { /inrht exch def } { pop } ifelse pop /inflag 1 def } bind def /Mouter { /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def inflag 1 eq { dup 0 lt { dup intop mul neg /yadtop exch def } if dup 0 gt { dup intop mul /yadbot 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0.0238095 0.47619 0.047619 0.47619 [ [ 0 0 0 0 ] [ 1 2 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 2 L 0 2 L closepath clip newpath .5 .1 .5 r .02381 1.95238 m .97619 1.95238 L .97619 1 L .5 1 L .5 1.47619 L .02381 1.47619 L .02381 1.95238 L F 0 g .5 Mabswid [ ] 0 setdash .02381 1.95238 m .97619 1.95238 L .97619 1 L .5 1 L .5 1.47619 L .02381 1.47619 L .02381 1.95238 L s .5 .1 .5 r .5 1 m .97619 1 L .97619 .04762 L .02381 .04762 L .02381 .52381 L .5 .52381 L .5 1 L F 0 g .5 1 m .97619 1 L .97619 .04762 L .02381 .04762 L .02381 .52381 L .5 .52381 L .5 1 L s % End of Graphics MathPictureEnd %%PSTrailer grestore pp_save restore end %%Trailer %%EOF --Apple-Mail-33-1064340819-- --Apple-Mail-26-1064340800 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; format=flowed Silvia Heubach On Feb 20, 2007, at 10:02 AM, Jeffrey Shallit wrote: > I am sorry, the version you sent me still does not work. In > particular, > it fails on the "includegraphics" command. Can you please send me a > version > that uses the psfig or epsfig packages instead. > > I am sorry to cause you more work. Are you using a version on a > macintosh? > It is my experience that the Mac version of tex does not work when > ported > to my Unix environment, unless one makes those kinds of changes. > > Jeffrey Shallit --Apple-Mail-26-1064340800--