########################################################################
##  RectTile.txt Save this file as  RectTile.txt  to use it,           #
# stay in the                                                          #
## same directory, get into Maple (by typing: maple <Enter> )          #
## and then type:  read RecTile.txt <Enter>                            #
## Then follow the instructions given there                            #
##                                                                     #
## Written by Pablo Blanco, Robert Dougherty-Bliss, Natalya Ter-Saakov #
#and                                                                   #
#Doron Zeilberger, Rutgers University ,                                #
##  DoronZeil at gmail dot com                                         # 
########################################################################


print(`First Written: April 2026: tested for Maple 2025 `):
print(`Version  : March 31, 2026  `):
print():
print(`This is RectTile.txt, A Maple package`):
print(`accompanying the article: `):
print(` In How Many Ways Can You Rectangle a Rectangle?`):
print(`By Pablo Blanco, Robert Dougherty-Bliss, Natalya Ter-Saakov and Doron Zeilberger`):
print():
print(`The most current version is available on WWW at:`):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/RectTile.txt .`):

print(`Please report all bugs to: DoronZeil at gmail dot com .`):
print():
print(`For general help, and a list of the MAIN functions,`):
print(` type "ezra();". For specific help type "ezra(procedure_name);" `):
print(`For a list of the supporting functions type: ezra1();`):
print(`For a list of the Paper functions type: ezraP();`):
print(`For a list of the Checking functions type: ezraC();`):
print():
 
with(plots):



ezraC:=proc()
if args=NULL then

print(`The Checking procedures are`):
print(`  CheckFmxt, CheckFmxw, CheckFmxw1w2`):

else
ezra(args):

fi:

end:

ezraP:=proc()
if args=NULL then

print(`The Papers procedures are`):
print(` PaperR, PaperRg, PaperRsv, PaperSg`):

else
ezra(args):

fi:

end:

ezra1:=proc()
if args=NULL then

print(`The SUPPORTING procedures are`):
print(` AdjMatW, AdjMatW1W2, Alpha1, Alpahbet1, amn, AsyAve, Bm, CrossSec, CtoR, Edges1, Edges11, KidsR, GuessRec, Grammar1, Grammar11, HV, KidsRp, KidsS, KidsSp, LD, Mm`):
print(`NewTileSi, NNT, PlotT, ShiftRi, TIr, TIrP, TIrPe, TIs, TIsP, TiToW`):

else
ezra(args):

fi:


end:

ezra:=proc()
if args=NULL then

print(` RectTile.txt: A Maple package for counting the number of ways of tiling rectangles by rectangles and by squares  `):

print(`The MAIN procedures are: Alphabet , DiG, DiGt, DiGw, DiGw1w2, Fmx, FmxPC, FmxSV, Fmxt, Fmxw,  Fmxw1w2, FmxwPC, GmxG, Grammar, RandRT, RandST, SeqR, SeqS  `):
print(``):


elif nargs=1 and args[1]=AdjMatW then
print(`AdjMatW(WG,w): Given a weighted directed graph G=[G1,WeightTable], and a variable name w, (in our format n:=nops(G1)=nops(WeightTable). Outputs the n by n weighted adjacency matrix in terms of powers of w. Try`):
print(`Try: AdjMatW(DiGw(2),w);`):

elif nargs=1 and args[1]=AdjMatW1W2  then
print(`AdjMatW1W2(G,w1,w2): Given a bi-weighted directed graph G=[G1,WeightBiTable], and a variable names w1, w2, (in our format n:=nops(G1)=nops(WeightTable). Outputs the n by n weighted adjacency matrix in terms of powers of w1 and w2. Try`):
print(` G:=DiGw1w2(2):AdjMatW1W2(G,w1,w2);`):

elif nargs=1 and args[1]=Alpha1 then
print(`Alpha1(m,n,i): the set of all letters that show up in the i-th place in Corpus(m,n); Try:`):
print(`Alpha1(3,4,1);`):

elif nargs=1 and args[1]=Alphabet then
print(`Alphabet(m): The starting letters, middle letters, and final letters of Corpus(m,n) for n>=2. Try:`):
print(`Alphabet(3);`):

elif nargs=1 and args[1]=Alphabet1 then
print(`Alphabet1(m,n): The starting letters, middle letters, and final letters of Corpus(m,n). Try: `):
print(`Alphabet1(3,4);`):

elif nargs=1 and args[1]=amn then
print(`amn(m,n): The number of ways to tile an m by n rectangle by rectangles, according to the Joshua Smith- Helena Verrill formula. Try:`):
print(`amn(4,10);`):

elif nargs=1 and args[1]=AsyAve then
print(`AsyAve(f,z,w,K): inputs a rational function f in z and w estiamtes [C1,C0] such that the coeff. of z^n of subs(w=1,diff(f,w)) divided by the coeff. of z^n  equals C1*n+C0 by looking`):
print(`at two consectutive terms around the coeff. of z^K. Try:`):
print(`AsyAve(1/(1-z*w-z^2),z,w,200);`):

elif nargs=1 and args[1]=Bm then
print(`Bm(m): the 2^(m-1) by 2^(m-1) matrix defined in the paper by Joshua Smith and Helena Verrill. Try:`):
print(`Bm(5);`):

elif nargs=1 and args[1]=CheckFmxt then
print(`CheckFmxt(m,N): checks the correctness of Fmxt(m,w,t) up to the coefficient of x^N. Try:`):
print(`CheckFmxt(2,5);`):


elif nargs=1 and args[1]=CheckFmxw then
print(`CheckFmxw(m,N): checks the correctness of Fmxw(m,x,w), up to the coefficient of x^N. Try:`):
print(`CheckFmxw(2,5);`):

elif nargs=1 and args[1]=CheckFmxw1w2 then
print(`CheckFmxw1w2(m,N): checks the correctness of Fmxw1w2(m,x,w1,w2), up to the coefficient of x^N. Try:`):
print(`CheckFmxw1w2(2,5);`):

elif nargs=1 and args[1]=Corners1 then
print(`Corners1(rec): given a rectangle with our convention [[i,j],[a,b]] denoting an a by b tile whose upper-right cell is (in matrix notation), [i,j], i.e. the upper right vertex is [i-1,j]`):
print(`outputs the four corners of the corresponindg a by by tile. It starts with the left-most bottom corner and proceeds, counterclockwise. Try:`):
print(`Corners1([[5,4],[2,3]]);`):

elif nargs=1 and args[1]=Corpus then
print(`Corpus(m,n): All the words for the tilings of an m by n rectangle by rectangles. Try:`):
print(`Corpus(3,4);`):

elif nargs=1 and args[1]=CrossSec then
print(`CrossSec(a): The set of edges in [0,a]x[0,1-], i,e, all the vertical edges in [0,a] and the horizontal edges coming out but not the edges of the vertices labeled [1,i] (0<=i<=a). Try:`):
print(`CrossSec(4);`):

elif nargs=1 and args[1]=CtoR then
print(`Taken from the Maple package Cfinie.txt`):
print(`CtoR(S,t), the rational function, in t, whose coefficients`):
print(`are the members of the C-finite sequence S. For example, try:`):
print(`CtoR([[1,1],[1,1]],t);`):

elif nargs=1 and args[1]=DiG then
print(`DiG(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m. Try:`):
print(`DiG(2);`):

elif nargs=1 and args[1]=DiGt then
print(`DiGt(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the "numtber of new tiles" weight of each vertex.  Try:`):
print(`DiGt(2);`):

elif nargs=1 and args[1]=DiGw then
print(`DiGw(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the weight of each vertex.  Try:`):
print(`DiGw(2);`):

elif nargs=1 and args[1]=DiGw1w2 then
print(`DiGw1w2(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the bi-weight of each vertex ([number of horizontal edges in the grid, number of vertical edges in the grid].`):
print(`Try:`):
print(`DiGw1w2(2);`):

elif nargs=1 and args[1]=Edges1 then
print(`Edges1(rec): given a rectangle with our convention [[i,j],[a,b]] denoting an a by b tile whose upper-right cell is. in matrix notation, `):
print(`outputs the set of edges in the perimeter of the rectangle rec. Try:`):
print(`Edges1([[1,4],[2,3]]);`):

elif nargs=1 and args[1]=Edges11 then
print(`Edges11(R): the set of edges in rectaongle R=[V1,V2,V3,V4] where V1 is the leftmost bottom corner and it is tranversed counterclockwise. Try:`):
print(`Edges11([[3,1],[3,3],[1,3],[1,1]]);`):

elif nargs=1 and args[1]=GmxG then
print(`GmxG(m,x,K): The generating function of SeqS(m,infinity); by (rigorous!) guessing using data up to K.  Try:`):
print(`GmxG(3,x,30);`):

elif nargs=1 and args[1]=Grammar then
print(`Grammar(m): the triple [StartingPairs, MiddlePairs,EndingPairs] in the grammar for Corpus(m,n) for all n>=3.`):
print(`Try:Grammar(3)`):

elif nargs=1 and args[1]=Grammar1 then
print(`Grammar1(m,n): the triple [StartingPairs, MiddlePairs,EndingPairs] in the grammar for Corpus(m,n), after checking that all the middle ones are the same. Try:`):
print(`Grammar1(3,4);`):

elif nargs=1 and args[1]=Grammar11 then
print(`Grammar11(m,n): the set of all the pairs of consecutive letters at the start, and then at every consecutive two columns and then at the end. Try`):
print(`Grammar11(3,5);`):

elif nargs=1 and args[1]=GuessRec then
print(`Taken from the Maple package Cfinie.txt`):
print(`GuessRec(L): inputs a sequence L and tries to guess`):
print(`a recurrence operator with constant cofficients `):
print(`satisfying it. It returns the initial  values and the operator`):
print(` as a list. For example try:`):
print(`GuessRec([1,1,1,1,1,1]);`):

elif nargs=1 and args[1]=HV then
print(`HV(S): Given a set of edges of P_m x P_n outputs the pair [h,v] where h is the number of horizontal edges, and v is the number of vertical edges. Try:`):
print(`HV({{[1,0],[2,0]}}); HV({{[0,1],[0,2]}});`):

elif nargs=1 and args[1]=KidsR then
print(`KidsR(L): all the configurations obtained from L by deciding the rectangular tile that belongs to the rightmost-upmost cell. Try:`):
print(`KidsR([4,4,4]);`):

elif nargs=1 and args[1]=KidsRp then
print(`KidsRp(L): all the pairs, [C,T] where C is configurations obtained from L by removing a legal rectangular tile T that belongs to the rightmost-upmost cell, that is also the North East Corner of T.`):
print(`A tile is represented by [RightUpCorner,[a,b]] where [a,b] are the dimensions of the tile. Try:`):
print(`KidsRp([4,4,4]);`):

elif nargs=1 and args[1]=KidsS then
print(`KidsS(L): all the configurations obtained from L by deciding the square tile that belongs to the rightmost-upmost cell. Try:`):
print(`KidsS([4,4,4]);`):

elif nargs=1 and args[1]=KidsSp then
print(`KidsSp(L): all the pairs, [C,T] where C is configurations obtained from L by removing a legal square tile T that belongs to the rightmost-upmost cell, that is also the North East Corner of T.`):
print(`A tile is represented by [RightUpCorner,[a,b]] where [a,b] are the dimensions of the tile. Try:`):
print(`KidsSp([4,4,4]);`):

elif nargs=1 and args[1]=LD then
print(`LD(L): inputs a list of pos. integers L describing a roullete outputs i with prob. proportioanl to L[i]. Try:`):
print(`LD([1,2,1]);`):

elif nargs=1 and args[1]=Mm then
print(`Mm(m): the 2^(m-1) by 2^(m-1) matrix defined in the paper by Joshua Smith and Helena Verrill. Try:`):
print(`Mm(5);`):

elif nargs=1 and args[1]=NewTileSi then
print(`NewTileSi(i,k): the signal of a new tile in a letter starting with horizontal edge {[i,0],[i,1]} and ending in horizontal edge {[i+k,0],[i+k,1]}, it outputs the`):
print(`set of edges that should be there followed by the set of horizontal edges that should not. Try:`):
print(`NewTileSi(2,4);`):

elif nargs=1 and args[1]=NNT then
print(`NNT(LE): The number of new tiles starting with the letter LE in Alphabet(m). Try:`):
print(`LE:=Alphabet(4)[2][1]; NNT(4,LE);`):

elif nargs=1 and args[1]=PaperR then
print(`PaperR(M,x): an article with the generating functions for tiling an m by n rectangle by RECTANGLES for m from 1 to M`):
print(`using the rigorous grammar. Try:`):
print(`PaperR(4,x);`):

elif nargs=1 and args[1]=PaperRg then
print(`PaperRg(M,x,K): an article with the generating functions for tiling an m by n rectangle by RECTANGLES for m from 1 to M`):
print(`using (rigorous!) guessing using up to K data points. Try:`):
print(`PaperRg(4,x,20);`):


elif nargs=1 and args[1]=PaperRsv then
print(`PaperRsv(M,x): an article with the generating functions for tiling an m by n rectangle by RECTANGLES for m from 1 to M`):
print(`using the Joshua Smith-Helena Verrill approach. Try:`):
print(`PaperRsv(4,x);`):

elif nargs=1 and args[1]=PaperSg then
print(`PaperSg(M,x,K): an article with the generating functions for tiling an m by n rectangle by  SQUARES for m from 1 to M`):
print(`using (rigorous!) guessing using up to K data points. Try:`):
print(`PaperSg(4,x,20);`):

elif nargs=1 and args[1]=PlotT then
print(`PlotT(a,b,S): plots the tiling T of [a]x[b]. Try:`):
print(`S:=RandRT([3,3,3,3,3]): PlotT(5,3,S);`):

elif nargs=1 and args[1]=RandRT then
print(`RandRT(L): a uniformaly-at-random tiling of the shape L by retangles. Try:`):
print(`RandRT([3,3,3]);`):

elif nargs=1 and args[1]=RandST then
print(`RandST(L): a uniformaly-at-random tiling of the shape L by squares. Try:`):
print(`RandST([3,3,3]);`):

elif nargs=1 and args[1]=SeqR then
print(`SeqR(k,N): The sequence of length N whose n-th term is the number of ways of tiling a k by (n-1) rectangle by rectangles. Try:`):
print(`SeqR(3,20);`):

elif nargs=1 and args[1]=Fmx then
print(`Fmx(m,x): The generating function of SeqR(m,infinity); Try:`):
print(`Fmx(3,x);`):

elif nargs=1 and args[1]=FmxG then
print(`FmxG(m,x,K): The generating function of SeqR(m,infinity); by (rigorous!) guessing using data up to K.  Try:`):
print(`FmxG(3,x,20);`):

elif nargs=1 and args[1]=FmxPC then
print(`FmxPC(m,x): For 1<=m<=9, the pre-computed value of Fmx(m,x). Try:`):
print(`FmxPC(5,x);`):

elif nargs=1 and args[1]=FmxSV then
print(`FmxSV(m,x): the same output as Fmx(m,x) (q.v.) but via the approach of Joshua Smith and Helena Verrill. Try:`):
print(`FmxSV(4,x);`):

elif nargs=1 and args[1]=Fmxt then
print(`Fmxt(m,x,t): The weighted generating function of SeqR(m,infinity), according to the weight t^NumberOfTiles. Try:`):
print(`Fmxt(3,x,t);`):

elif nargs=1 and args[1]=Fmxw then
print(`Fmxw(m,x,w): The weighted generating function of SeqR(m,infinity); Try:`):
print(`Fmxw(3,x,w);`):

elif nargs=1 and args[1]=Fmxw1w2 then
print(`Fmxw1w2(m,x,w1,w2): The bi-weighted generating function of SeqR(m,infinity), according to the weight:`):
print(`w1^NumberOfHorizontalEdges*w2^NumberOfVerticalEdges`):
print(`Try:`):
print(`Fmxw1w2(3,x,w1,w2);`):

elif nargs=1 and args[1]=FmxwPC then
print(`FmxwPC(m,x,w): The pre-computed value of the bi-variate generating  function of SeqR(m,infinity); For n from 1 to 6. Try:`):
print(`FmxwPC(3,x,w);`):

elif nargs=1 and args[1]=SeqS then
print(`SeqS(k,N): The sequence of length N whose n-th term is the number of ways of tiling a k by (n-1) rectangle by squares. Try:`):
print(`SeqS(3,20);`):

elif nargs=1 and args[1]=FmxG then
print(`FmxG(m,x,K): The generating function of SeqS(m,infinity); by (rigorous!) guessing using data up to K.  Try:`):
print(`FmxG(3,x,20);`):

elif nargs=1 and args[1]=ShiftRi then
print(`ShiftRi(C,i): adding [i,0] to each vertex of each edgde of C.`):

elif nargs=1 and args[1]=TIr then
print(`TIr(L): The number of ways to tile the configuration L with rectangular tiles. Try:`):
print(`TIr([3,3,3]);`):

elif nargs=1 and args[1]=TIrP then
print(`TIrP(L): The set of tilings of  the configuration L with rectangular tiles. Try:`):
print(`TIrP([3,3,3]);`):

elif nargs=1 and args[1]=TIrPe then
print(`TIrPe(L): The set of tilings of  the configuration L with rectangular tiles in terms of the set of edges partitipating in the borders of the tiles. Try:`):
print(`TIrPe([3,3,3]);`):

elif nargs=1 and args[1]=TIs then
print(`TIs(L): The number of ways to tile the configuration L with square tiles. Try:`):
print(`TIs([3,3,3]);`):

elif nargs=1 and args[1]=TIsP then
print(`TIsP(L): The set of tilings of  the configuration L with square tiles. Try:`):
print(`TIsP([3,3,3]);`):

elif nargs=1 and args[1]=TiToW then
print(`TiToW(m,n,T): The word corresponding to the tiling T on an m by n board`):

else
 print(`There is no such thing as`, args):

fi:


end:


#KidsR(L): all the configurations obtained from L by deciding the rectangular tile that belongs to the rightmost-upmost cell. Try:
#KidsR([4,4,4]);
KidsR:=proc(L) local k,i,j,j1,ka,gu,j11:
k:=nops(L):
if L=[0$k] then
 RETURN(FAIL):
 fi:
i:=max[index](L):

for j from i+1 to k while L[j]=L[i] do od:
j:=j-1:

gu:={}:
for j1 from i to j do
 gu:=gu union {seq([op(1..i-1,L),seq(L[j11]-ka,j11=i..j1),op(j1+1..k,L)],ka=1..L[i])}:
od:

gu:

end:

#TIr(L): The number of ways to tile the configuration L with rectangular tiles. Try:
#TIr([3,3,3]);
TIr:=proc(L) local k,gu,gu1:
option remember:
k:=nops(L):
if L=[0$k] then
 RETURN(1):
fi:

gu:=KidsR(L):

add(TIr(gu1),gu1 in gu):
end:


#SeqR(k,N): The sequence of length N+1 whose n-th term is the number of ways of tiling a k by (n-1) rectangle by rectangles. Try:
#SeqR(4,20);
SeqR:=proc(k,N) local i:
[seq(TIr([i$k]),i=0..N)]:
end:


#KidsS(L): all the configurations obtained from L by deciding the square tile that belongs to the rightmost-upmost cell. Try:
#KidsS([4,4,4]);
KidsS:=proc(L) local k,i,j,j1,gu,j11:
k:=nops(L):
if L=[0$k] then
 RETURN(FAIL):
 fi:
i:=max[index](L):

for j from i+1 to k while L[j]=L[i] do od:
j:=j-1:

gu:={}:
for j1 from i to min(j,L[i]+i-1) do
 gu:=gu union {[op(1..i-1,L),seq(L[j11]-(j1-i+1),j11=i..j1),op(j1+1..k,L)]}:
od:

gu:

end:



#TIs(L): The number of ways to tile the configuration L with square tiles. Try:
#TIs([3,3,3]);
TIs:=proc(L) local k,gu,gu1:
option remember:
k:=nops(L):
if L=[0$k] then
 RETURN(1):
fi:

gu:=KidsS(L):

add(TIs(gu1),gu1 in gu):
end:


#SeqS(k,N): The sequence of length N+1 whose n-th term is the number of ways of tiling a k by (n-1) rectangle by squares. Try:
#SeqS(4,20);
SeqS:=proc(k,N) local i:
[seq(TIs([i$k]),i=0..N)]:
end:



#KidsRp(L): all the pairs, [C,T] where C is configurations obtained from L by removing a legal rectangular tile T that belongs to the rightmost-upmost cell, that is also the North East Corner of T.
#A tile is represented by [RightUpCorner,[a,b]] where [a,b] are the dimensions of the tile. Try:
#KidsRp([4,4,4]);
KidsRp:=proc(L) local k,i,j,j1,ka,gu,j11,L1,T1:
k:=nops(L):
if L=[0$k] then
 RETURN(FAIL):
 fi:
i:=max[index](L):

for j from i+1 to k while L[j]=L[i] do od:
j:=j-1:

gu:={}:

for j1 from i to j do
 for ka from 1 to L[i] do
 L1:=[op(1..i-1,L),seq(L[j11]-ka,j11=i..j1),op(j1+1..k,L)]:
 T1:=[[i,L[i]],[j1-i+1,ka]]:
 gu:=gu union {[L1,T1]}:
od:
od:

gu:

end:





#TIrP(L): The set of tilings of  the configuration L with rectangular tiles. Try:
#TIrP([3,3,3]);
TIrP:=proc(L) local k,mu,mu1,lu,lu1,gu:
option remember:
k:=nops(L):
if L=[0$k] then
 RETURN({{}}):
fi:


mu:=KidsRp(L):

gu:={}:

for mu1 in mu do
lu:=TIrP(mu1[1]):
gu:=gu union {seq(lu1 union {mu1[2]} , lu1 in lu)}:
od:
gu:
end:








#KidsSp(L): all the pairs, [C,T] where C is configurations obtained from L by removing a legal square tile T that belongs to the rightmost-upmost cell, that is also the North East Corner of T.
#A tile is represented by [RightUpCorner,[a,b]] where [a,a] are the dimensions of the tile. Try:
#KidsSp([4,4,4]);
KidsSp:=proc(L) local mu,mu1,gu:
mu:=KidsRp(L):

gu:={}:

for mu1 in mu do
if mu1[2][2][1]=mu1[2][2][2] then
 gu:=gu union {mu1}:
fi:
od:
gu:

end:



#TIsP(L): The set of tilings of  the configuration L with square tiles. Try:
#TIsP([3,3,3]);
TIsP:=proc(L) local k,mu,mu1,lu,lu1,gu:
option remember:
k:=nops(L):
if L=[0$k] then
 RETURN({{}}):
fi:


mu:=KidsSp(L):

gu:={}:

for mu1 in mu do
lu:=TIsP(mu1[1]):
gu:=gu union {seq(lu1 union {mu1[2]} , lu1 in lu)}:
od:
gu:
end:

#LD(L): inputs a list of pos. integers L describing a roullete outputs i with prob. proportioanl to L[i]. Try:
#LD([1,2,1]);
LD:=proc(L) local t,ra,i,j:
t:=convert(L,`+`):
ra:=rand(1..t)():

for i from 1 while add(L[j],j=1..i)<ra do od:
i:
end:


#RandRT(L): a uniformaly-at-random tiling of the shape L by retangles. Try:
#RandRT([3,3,3]);
RandRT:=proc(L) local gu,i1,i,L1,T1:
if L=[0$nops(L)] then
 RETURN({}):
fi:

gu:=convert(KidsRp(L),list):
i:=LD([seq(TIr(gu[i1][1]),i1=1..nops(gu))]):
L1:=gu[i][1]:
T1:=gu[i][2]:

RandRT(L1) union {T1}:

end:


#LD(L): inputs a list of pos. integers L describing a roullete outputs i with prob. proportioanl to L[i]. Try:
#LD([1,2,1]);
LD:=proc(L) local t,ra,i,j:
t:=convert(L,`+`):
ra:=rand(1..t)():

for i from 1 while add(L[j],j=1..i)<ra do od:
i:
end:


#RandST(L): a uniformaly-at-random tiling of the shape L by squares. Try:
#RandST([3,3,3]);
RandST:=proc(L) local gu,i1,i,L1,T1:
if L=[0$nops(L)] then
 RETURN({}):
fi:

gu:=convert(KidsSp(L),list):
i:=LD([seq(TIs(gu[i1][1]),i1=1..nops(gu))]):
L1:=gu[i][1]:
T1:=gu[i][2]:

RandST(L1) union {T1}:

end:





#PlotT(a,b,S): plots the tiling T of [a]x[b]. Try:

PlotT:=proc(a,b,S) local p,i,j,s,pt,rec,Pt:

p:=plot([[0,0],[b,0],[b,a],[0,a],[0,0]],thickness=2,axes=none,scaling=constrained):


for i from 1 to a-1 do
p:=p,plot([[0,i],[b,i]]):
od:
for j from 1 to b-1 do
p:=p,plot([[j,0],[j,a]]):
od:


for s in S do
pt:=s[1]:
rec:=s[2]:

Pt:=[pt[2],a-pt[1]+1]:

p:=p,plot([Pt,Pt-[rec[2],0],Pt-[rec[2],rec[1]], Pt-[0,rec[1]],Pt], thickness=5):

od:


display(p):
end:





#CrossSec(a): The set of edges in [0,a]x[0,1-], i,e, all the vertical edges in [0,a] and the horizontal edges coming out but not the edges of the vertices labeled [1,i] (0<=i<=a). Try:
#CrossSec(4);
CrossSec:=proc(a) local i:
{seq({[i,0],[i+1,0]},i=0..a-1), seq({[i,0],[i,1]},i=0..a)}:
end:



#ShiftRi(C,i): adding [i,0] to each vertex of each edgde of C. 
ShiftRi:=proc(C,i) local e,v:
{seq({seq([0,i]+v,v in e)},e in C)}:
end:






#TIrPe(L): The set of tilings of  the configuration L with rectangular tiles expressed in terms of the set of edges in the border of the tiles. Try:
#TIrPe([3,3,3]);
TIrPe:=proc(L) local gu,gu1,gu11,S:
gu:=TIrP(L):
S:={seq({seq(op(Edges1(gu11)),gu11 in gu1)},gu1 in gu)}:
end:


#TiToW(m,n,T): The word corresponding to the tiling T on an m by n board
TiToW:=proc(m,n,T) local lu,i:
lu:=CrossSec(m):
[seq(ShiftRi(ShiftRi(lu,i) intersect T,-i),i=0..n)]:
end:



#Corpus(m,n): All the words for the tilings of an m by n rectangle by rectangles. Try:
#Corpus(3,4);
Corpus:=proc(m,n) local gu,T:
option remember:
gu:=TIrPe([n$m]):
{seq(TiToW(m,n,T), T in gu)}:
end:




#Alpha1(m,n,i): the set of all letters that show up in the i-th place in Corpus(m,n); Try:
#Alpha1(3,4,1);
Alpha1:=proc(m,n,i) local gu,gu1:
option remember:
gu:=Corpus(m,n):
{seq(gu1[i],gu1 in gu)}:
end:

#Alphabet1(m,n): The starting letters, middle letters, and final letters of Corpus(m,n). Try: Alphabet(3,4);
Alphabet1:=proc(m,n) local gu1,gu2,gu3,i:
gu1:=Alpha1(m,n,1):
gu2:=Alpha1(m,n,2):

for i from 3 to n do

if Alpha1(m,n,i)<>gu2 then
print(`The letters that show up in the`, i, ` place are`):
print(Alpha1(m,n,i)):
   RETURN(FAIL):
 fi:
 od:
 gu3:=Alpha1(m,n,n+1):

[gu1,gu2,gu3]:

end:



#Alphabet(m): The starting letters, middle letters, and final letters of Corpus(m,n) for n>=2. Try:
#lphabet(3);
Alphabet:=proc(m):
Alphabet1(m,3):
end:


#Grammar11(m,n): the set of all the pairs of consecutive letters at the start, and then at every consecutive two columns and then at the end
Grammar11:=proc(m,n) local gu,ST,i,gu1:
gu:=Corpus(m,n):

for i from 1 to n do
ST[i]:={seq([gu1[i],gu1[i+1]],gu1 in gu)}:
od:

[seq(ST[i],i=1..n)]:


end:



#Corners1(rec): given a rectangle with our convention [[i,j],[a,b]] denoting an a by b tile whose upper-right cell is (in matrix notation), [i,j], i.e. the upper right vertex is [i-1,j]
#outputs the four corners of the corresponindg a by by tile. It starts with the left-most bottom corner and proceeds, counterclockwise. Try:
#Corners1([[5,4],[2,3]]);
Corners1:=proc(rec) local i,j,a,b:
i:=rec[1][1]: j:=rec[1][2]:
a:=rec[2][1]: b:=rec[2][2]:
[[i-1+a,j-b],[i-1+a,j],[i-1,j],[i-1,j-b]]:
end:

#Edges11(R): the set of edges in rectaongle R=[V1,V2,V3,V4] where V1 is the leftmost bottom corner and it is tranversed counterclockwise. Try:
#Edges11([[3,1],[3,3],[1,3],[1,1]]);
Edges11:=proc(R) local V1,V2,V3,V4,i1,j1,gu:
V1:=R[1]:V2:=R[2]:V3:=R[3]: V4:=R[4]:

if not (V1[1]=V2[1] and V1[2]<V2[2]) then
  RETURN(FAIL):
  else
  gu:={seq({[V1[1],j1], [V1[1],j1+1]},j1=V1[2]..V2[2]-1)}:
fi:


if not (V4[1]=V3[1] and V4[2]<V3[2]) then
  RETURN(FAIL):
  else
  gu:=gu union {seq({[V4[1],j1], [V4[1],j1+1]},j1=V4[2]..V3[2]-1)}:
fi:


if not (V3[2]=V2[2] and V3[1]<V2[1]) then
  RETURN(FAIL):
  else
  gu:=gu union {seq({[i1,V3[2]], [i1+1,V2[2]]},i1=V3[1]..V2[1]-1)}:
fi:

if not (V4[2]=V1[2] and V4[1]<V1[1]) then
  RETURN(FAIL):
  else
  gu:=gu union {seq({[i1,V4[2]], [i1+1,V1[2]]},i1=V4[1]..V1[1]-1)}:
fi:


gu:

end:



#Edges1(rec): given a rectangle with our convention [[i,j],[a,b]] outputs the set of edges on its perimeter
#Edges1([[5,4],[2,3]]);
Edges1:=proc(rec):
Edges11(Corners1(rec)):
end:




#Grammar1(m,n): the triple [StartingPairs, MiddlePairs,EndingPairs] in the grammar for Corpus(m,n), after checking that all the middle ones are the same. Try:
#Grammar1(3,4);
Grammar1:=proc(m,n) local gu,i:

gu:=Grammar11(m,n):

if nops({seq(gu[i],i=2..n-1)})<>1 then
    RETURN(FAIL):
fi:

[gu[1],gu[2],gu[n]]:

end:


#Grammar(m): the triple [StartingPairs, MiddlePairs,EndingPairs] in the grammar for Corpus(m,n) for all n>=3.
#Try:Grammar(3)
Grammar:=proc(m):
Grammar1(m,3):
end:


#DiG(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m. Try:
#DiG(2);
DiG:=proc(m) local A,G,T,a,A2,Na,i,a1,g:
A:=Alphabet(m):

G:=Grammar(m):

if A[1] minus A[2]<>{} then
 RETURN(FAIL):
fi:
if nops(A[3])<>1 then
 RETURN(FAIL):
fi:

a:=nops(A[2]):
A2:=convert(A[2],list):

Na[A[3][1]]:=a+2:

for i from 1 to a do
 Na[A2[i]]:=i+1:
od:

T[1]:={seq(Na[a1], a1 in A[1])}:

for i from 2 to a+1 do
 T[i]:={}:
od:

T[a+2]:={}:

for g in G[2] union G[3] do
 T[Na[g[1]] ]:=  T[Na[g[1]] ] union {Na[g[2]]}:
od:

[seq(T[i],i=1..a+2)]:

end:



with(linalg):

#AdjMat(G): Given a directed graph G in our format n:=nops(G). Outputs the n by n adjacency matrix
#Try:G:=RDG(5,1/2);AdjMat(G);

AdjMat:=proc(G) local M,M1,i,n,j:
n:=nops(G):
M:=[]:

for i from 1 to n do
M1:=[]:
for j from 1 to n do
 if member(j, G[i]) then
   M1:=[op(M1),1]:
 else
   M1:=[op(M1),0]:
fi:
od:
M:=[op(M),M1]:
od:
M:
end:


#WalkGF(G,x): Inputs a directed graph G=[V,T], where T[v] is the set of outgoing neighbors of v, and a variable x, outputs the matrix of rational functions whose (i,j) entry is
#the generating function in x, according to lengths of walks from i to j. Try:
#G:=RDG(5,1/2); WalkGF(G,x);
WalkGF:=proc(G,x) local n,M,i:
n:=nops(G):
M:=AdjMat(G):
M:=matrix([seq(expand([0$(i-1),1,0$(n-i)]-x*M[i]),i=1..n)]):
normal(inverse(M))[1,n]:
end:



#Fmx(m,x): The generating function of SeqR(m,infinity); Try:
#Fmx(3,x);
Fmx:=proc(m,x) local G:
G:=DiG(m):
normal(1+WalkGF(G,x)/x):
end:


#FmxPC(m,x): the pre-computed output for Fmx(m,x) for m from 1 to 5
FmxPC:=proc(m,x) local L:
option remember:
if not (m>=1 and m<=9) then
 print(m, `should have been between 1 and 9`):
 RETURN(FAIL):
fi:

L:=[(x-1)/(2*x-1),(3*x^2-4*x+1)/(7*x^2-6*x+1),(19*x^3-29*x^2+11*x-1)/(51*x^3-55*x^2+15*x-1),
(3832*x^6-8492*x^5+6722*x^4-2468*x^3+441*x^2-36*x+1)/(11680*x^6-20980*x^5+13840*x^4-4280*x^3+645*x^2-44*x+1),
(39672144*x^10-110891556*x^9+124284414*x^8-74544838*x^7+26669637*x^6-5961522*x^5+841659*x^4-73608*x^3+3769*x^2-100*x+1)/(135762480*x^10-326041524*x^9+320708934*x^8-170972730*x^7+54776249*x^6-11002298*x^5+1395665*x^4-109292*x^3+4975*x^2-116*x+1),
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9+474130818860955757*x^8-8428505768539143*x^7+121156186144349*x^6-1376674485693
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0274*x^71+138712432168636454534029423154247580873361993987345458046300902205822\
68188552007043984761959903427470913562744779844315519451265751838495673732*x^70
-762716338855203848463297987105271616564363679149944934112467368259861014693283\
583615476723661737803081755605684308217810800806017384295406820*x^69+4032722702\
5995623921057053895757690561202356638492792986876876719464695290490932973116067\
781185943347851175721924294303622419946383891126563*x^68-2050251865983980295481\
9670578418914766445109926048241916254648222947738363112537356522641835222579426\
47703635680893586824748189104532565924*x^67+10022344399359375774172592016762457\
1003254934375954142611864851804522785126552456200567230639106130101205313460135\
819058419455185989973104*x^66-4710404119977141076160266162284232991779794598930\
7172767820094748748570913571078211186463931157879740460875482272525182080113114\
10900712*x^65+21283398923408360010036131998815033344281724346128972486749721646\
7953378608443662366265496784938317998025846511451744308932620849379822*x^64-924\
4399277407397398864597293575164108614775376720657110576139869674081302577822600\
371007275599258850889009851857306114153740053223124*x^63+3859453720458495078131\
6507986494468491771568918101450714562925515115400718583839705288541989716083204\
6629458246691721463991136235570*x^62-154856873379574604524034563475077522280009\
0436367887304111814612520961758852239950622532637705785495054719308465107260773\
2032134750*x^61+597082228388421026195149248372800537941910637795629899190013827\
897523529718753786048738844258535391236621807978580646123281714448*x^60-2211931\
2633555259353219442820179068832333545847369499134227030831020494561252190802924\
217708097617175087534089330566411721379048*x^59+7871743216335526312042714393054\
8371957520284521660649095442332697421589691070804606213900937039287264713170364\
3115295762934842*x^58-269062406228313596102892369851914830813304748623730719635\
72557699266782577576541210220941822882918678924494591210474909383076*x^57+88314\
4519095283624021367437420072782403977468010562194308579499046735197743355451696\
731425762367662434831174410769079369556*x^56-2783002974504963694983680866025880\
3134045395988259440784351962910965298224975703748081318236948929890189455571341\
504652944*x^55+8417780650891685355898126551895964363031521448173114486307555228\
65139843373331242927378643721168740945545431070587657972*x^54-24432880157971282\
0241158534306718933022571360340638218046005322818031627555977970435366849746471\
68782024281714636545408*x^53+68034287962775441231212482447870126679411402389907\
2787892674272497581331919604379452379560272957432501784809846666798*x^52-181690\
3024809058634403993324194706674079848151991926516899808326009194859029106484423\
7499362454585631756966427500818*x^51+465213457129330748920747282176227708939436\
172283565293422877875857241258558840793808058979227422796883013718421434*x^50-\
1141684819195366498043468083287304709200746670077019301527571065034486753590429\
4498421019076129052601316133000482*x^49+268448494566819859409699135791641302662\
432589568035477919283889807551781822833261761184857058295379912818761248*x^48-\
6045524178736100654289262701610937734376408295532185747413286519120190318451210\
575132268607850232824928894872*x^47+1303443696967840512819000789404051905351433\
62799942852049548581054213613016584309205951601708574245695949618*x^46-26893795\
7513724771573387573451628705668738856239806133201514031220650586107370815283272\
5916753820408898638*x^45+530784416520939667652855982268793496553942231154158188\
91145774474706727500182387617264787581004775242577*x^44-10015741168509147612659\
5363007194132328446676441040696835031373870243861299268455494944575816409017151\
2*x^43+180603740105876057824559733209468871935108721780604856017560210454997682\
11989002261929557336804524040*x^42-31103940717503405675559208985821772116467546\
0667455834787082595085103256429484825162637145022581508*x^41+511329741306434119\
0261266360919595207736850753636971768616004965096142849420018726145702462171112
*x^40-8019001426740620280751279299314446525320386636165936774834504534932104997\
3662020707275141563334*x^39+119892893644910363632146979079337824757087058757211\
4871513979620387915846010823918429451328344*x^38-170774639259867172986057496068\
87303556515273919374846511048309829734169419925472478708341108*x^37+23157638041\
7238776778657262784476593725976396822238279929951374791529795451912405615810969
*x^36-2987240370448894626874210952934075093368206577026373623208136445458210491\
520852403627110*x^35+3662634035823912044738179957234506612074694767277835696838\
3781032324509090194562914418*x^34-426464980807819645792473429459312751201341008\
790297742493102064751858295820005000806*x^33+4711220463698449303216897732978603\
194693250934550292637518877167106766737699826184*x^32-4932986020885656346643412\
0916797319439025679285730883041416340683046997745607964*x^31+
489045160120843007258878422687536628296384866229991659900218011527535194995326*
x^30-\
4585156025935507194761431756525801117643417953605040091961465751520490879038*x^
29+40606247407895833048728319436884292269297004774397075119293853945451687279*x
^28-339229130344222214790000385531984709710646882031274017003145523160003048*x^
27+2669557251456657785341755588783847311706084331320721137093648141458712*x^26-\
19759091975943317500268719023594030849294492538773286083100324146762*x^25+
137327731845583907170218899523118732077537079387705169163317163786*x^24-\
894611685756358155813133794754864838609124072919996421891586032*x^23+
5451922321990453357938799920675229965494735249866882700809062*x^22-\
31015631966700151234208381779381630186570786439277090206290*x^21+
164330275288112917461766486039451921246825654009748654509*x^20-\
808818645492787660660690701920760591772803252958302156*x^19+
3687717303548996333132314190491278485525749708886268*x^18-\
15526756641130818040995038219332878692260613069886*x^17+
60160240524568681987049431148156732128143045146*x^16-\
213673589168070501662429666304112066245566190*x^15+
692619241574427848455676784706283271824988*x^14-\
2038777814369390252423441154237944993562*x^13+
5418618378467804326177666257742117147*x^12-12917247926039167494881285272579058*
x^11+27405709205119485419355946682408*x^10-51274082375249535040034193152*x^9+
83658584800484711753101191*x^8-117418382077141470947474*x^7+
139341721707907868306*x^6-136708793010911560*x^5+107562650496809*x^4-\
64966624472*x^3+28131602*x^2-7722*x+1)]:
L[m]:
end:


###From Cfinite.txt

#SeqFromRec(S,N): Inputs S=[INI,ope]
#where INI is the list of initial conditions, a ope a list of
#size L, say, and a recurrence operator ope, codes a list of
#size L, finds the first N0 terms of the sequence satisfying
#the recurrence f(n)=ope[1]*f(n-1)+...ope[L]*f(n-L).
#For example, for the first 20 Fibonacci numbers,
#try: SeqFromRec([[1,1],[1,1]],20);
SeqFromRec:=proc(S,N) local gu,L,n,i,INI,ope:
INI:=S[1]:ope:=S[2]:
if not type(INI,list) or not type(ope,list) then
 print(`The first two arguments must be lists `):
 RETURN(FAIL):
fi:
L:=nops(INI):
if nops(ope)<>L then
 print(`The first two arguments must be lists of the same size`):
RETURN(FAIL):
fi:

if not type(N,integer) then
 print(`The third argument must be an integer`, L):
 RETURN(FAIL):
fi:

if N<L then
 RETURN([op(1..N,INI)]):
fi:

gu:=INI:

for n from 1 to N-L do
 gu:=[op(gu),expand(add(gu[nops(gu)-i+1]*ope[i],i=1..L))]:
od:

gu:
end:



#GuessRec1(L,d): inputs a sequence L and tries to guess
#a recurrence operator with constant cofficients of order d
#satisfying it. It returns the initial d values and the operator
# as a list. For example try:
#GuessRec1([1,1,1,1,1,1],1);
GuessRec1:=proc(L,d) local eq,var,a,i,n:

if nops(L)<=2*d+2 then
 print(`The list must be of size >=`, 2*d+3 ):
 RETURN(FAIL):
fi:

var:={seq(a[i],i=1..d)}:

eq:={seq(L[n]-add(a[i]*L[n-i],i=1..d),n=d+1..nops(L))}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
else
 RETURN([[op(1..d,L)],[seq(subs(var,a[i]),i=1..d)]]):
fi:

end:



#GuessRec(L): inputs a sequence L and tries to guess
#a recurrence operator with constant cofficients 
#satisfying it. It returns the initial  values and the operator
# as a list. For example try:
#GuessRec([1,1,1,1,1,1]);
GuessRec:=proc(L) local gu,d:

for d from 1 to trunc(nops(L)/2)-2 do
 gu:=GuessRec1(L,d):
 if gu<>FAIL then
   RETURN(gu):
 fi:
od:
FAIL:

end:


#CtoR(S,t), the rational function, in t, whose coefficients
#are the members of the C-finite sequence S. For example, try:
#CtoR([[1,1],[1,1]],t);

CtoR:=proc(S,t) local D1,i,N1,L1,f,f1,L:
if not (type(S,list) and  nops(S)=2 and type(S[1],list) and type(S[2],list)
        and nops(S[1])=nops(S[2]) and type(t, symbol) ) then
   print(`Bad input`):
   RETURN(FAIL):
fi:

D1:=1-add(S[2][i]*t^i,i=1..nops(S[2])):
N1:=add(S[1][i]*t^(i-1),i=1..nops(S[1])):
L1:=expand(D1*N1):


L1:=add(coeff(L1,t,i)*t^i,i=0..nops(S[1])-1):
f:=L1/D1:
L:=degree(D1,t)+10:
f1:=taylor(f,t=0,L+1):
if expand([seq(coeff(f1,t,i),i=0..L)])<>expand(SeqFromRec(S,L+1)) then
print([seq(coeff(f1,t,i),i=0..L)],SeqFromRec(S,L+1)):
 RETURN(FAIL):
else
 RETURN(f):
fi:
end:

###End From Cfinite.txt



#FmxG(m,x,K): The generating function of SeqR(m,infinity); by (rigorous!) guessing using data up to K.  Try:
#FmxG(3,x,20);
FmxG:=proc(m,x,K) local L,K1:

L:=GuessRec(SeqR(m,10)):
if L<>FAIL then
 RETURN(CtoR(L,x)):
fi:

for K1 from 15 by 5 to K do
L:=GuessRec(SeqR(m,K1)):
if L<>FAIL then
 RETURN(CtoR(L,x)):
fi:
od:
FAIL:

end:

#GmxG(m,x,K): The generating function of SeqS(m,infinity); by (rigorous!) guessing using data up to K.  Try:
#GmxG(3,x,30);
GmxG:=proc(m,x,K) local L,K1:

L:=GuessRec(SeqS(m,10)):
if L<>FAIL then
 RETURN(CtoR(L,x)):
fi:

for K1 from 15 by 5 to K do
L:=GuessRec(SeqS(m,K1)):
if L<>FAIL then
 RETURN(CtoR(L,x)):
fi:
od:
FAIL:

end:


#PaperR(M,x): an article with the generating functions for tiling an m by n rectangle by rectangles for m from 1 to M
#using the rigorous grammar. Try:
#PaperR(4,x);
PaperR:=proc(M,x) local m,gu,n,f,i:

print(`Generating functions for the number of ways to tile an m by n rectangle with Rectangular Tiles of any Size`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for m from 1 to M do
f:=Fmx(m,x):
print(``):
print(`Theorem number`, m):
print(``):
print(`Let `, a[m](n), `be the number of ways to tile an`, m, `by n rectangle with RECTANGULAR tiles , then `):
print(``):
print(Sum(a[m](n)*x^n,n=0..infinity)=f):
print(``):
print(`and in Maple notation`):
print(``):
lprint(f):
print(``):

f:=taylor(f,x=0,33):
gu:=[seq(coeff(f,x,i),i=0..30)]:
print(`The first 31 terms of the sequence, starting at n=0 are `):
lprint(gu):

od:

print(``):
print(`----------------------`):
print(``):
print(`This ends this article that took`, time(), `seconds to generate `):
print(``):
end:



#PaperRg(M,x,K): an article with the generating functions for tiling an m by n rectangle by rectangles for m from 1 to M
#using (rigorous) guessing by fitting data up to K data points. Try:
#PaperRg(4,x,30);
PaperRg:=proc(M,x,K) local m,gu,n,f,i:

print(`Generating functions for the number of ways to tile an m by n rectangle with Rectangular Tiles of any Size`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for m from 1 to M do
f:=FmxG(m,x,K):
if f<>FAIL then
print(``):
print(`Theorem number`, m):
print(``):
print(`Let `, a[m](n), `be the number of ways to tile an`, m, `by n rectangle with RECTANGULAR tiles , then `):
print(``):
print(Sum(a[m](n)*x^n,n=0..infinity)=f):
print(``):
print(`and in Maple notation`):
print(``):
lprint(f):
print(``):

f:=taylor(f,x=0,33):
gu:=[seq(coeff(f,x,i),i=0..30)]:
print(`The first 31 terms of the sequence, starting at n=0 are `):
lprint(gu):
fi:

od:

print(``):
print(`----------------------`):
print(``):
print(`This ends this article that took`, time(), `seconds to generate `):
print(``):
end:



#PaperSg(M,x,K): an article with the generating functions for tiling an m by n rectangle by squares for m from 1 to M
#using (rigorous) guessing by fitting data up to K data points. Try:
#PaperSg(4,x,30);
PaperSg:=proc(M,x,K) local m,gu,n,f,i:

print(`Generating functions for the number of ways to tile an m by n rectangle with Square Tiles of any Size`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for m from 1 to M do
f:=GmxG(m,x,K):
if f<>FAIL then
print(``):
print(`Theorem number`, m):
print(``):
print(`Let `, a[m](n), `be the number of ways to tile an`, m, `by n rectangle with SQUARE tiles , then `):
print(``):
print(Sum(a[m](n)*x^n,n=0..infinity)=f):
print(``):
print(`and in Maple notation`):
print(``):
lprint(f):
print(``):

f:=taylor(f,x=0,33):
gu:=[seq(coeff(f,x,i),i=0..30)]:
print(`The first 31 terms of the sequence, starting at n=0 are `):
lprint(gu):
fi:

od:

print(``):
print(`----------------------`):
print(``):
print(`This ends this article that took`, time(), `seconds to generate `):
print(``):
end:




#Start the Smith-Verrill arroach


#Mm(m): The matrix M_m(m) in the paper "On dividing rectangles into rectangles" bu Joshua Smith and Helena Verrill
Mm:=proc(m) local i,j,gu11,gu12,gu21,gu22:
option remember:
if m=1 then
 RETURN([[2]]):
 else
 gu11:=Mm(m-1):
 gu22:=[seq([seq(2*gu11[i][j],j=1..nops(gu11))],i=1..nops(gu11))]:
 gu12:=Bm(m-1):
   gu21:=Bm(m-1):
   
RETURN([seq([op(gu11[i]),op(gu12[i])],i=1..nops(gu11)),
        seq([op(gu21[i]),op(gu22[i])],i=1..nops(gu21))]):
fi:

end:

#Bm(m): The matrix B_m(m) in the paper "On dividing rectangles into rectangles" bu Joshua Smith and Helena Verrill
Bm:=proc(m) local i,gu11,gu12,gu21,gu22:
option remember:
if m=1 then
 RETURN([[1]]):
 else
 gu11:=Bm(m-1):
 gu12:=Bm(m-1):
 gu21:=Bm(m-1):
  gu22:=Mm(m-1):
   
RETURN([seq([op(gu11[i]),op(gu12[i])],i=1..nops(gu11)),
        seq([op(gu21[i]),op(gu22[i])],i=1..nops(gu21))]):
fi:

end:

with(linalg):

#amn(m,n): The number of ways to fild an m by n rectangle according to the Smith-Verrill formula
amn:=proc(m,n) local M,v,vT:
M:=convert(Mm(m),matrix):
M:=evalm(M&^(n-1)):
v:=convert([[1$2^(m-1)]],matrix):
vT:=convert([[1]$2^(m-1)],matrix):
multiply(multiply(v,M),vT)[1,1]:
end:

#FmxSV(m,x): the same output as Fmx (q.v.) but via the approach of Joshua Smith and Helena Verrill. Try:
#FmxSV(4,x);
FmxSV:=proc(m,x) local M,v,vT,i:
M:=Mm(m):
M:=expand([seq( [ 0$(i-1),1,0$(2^(m-1)-i)]-x*M[i],   i=1..2^(m-1) )]):
M:=matrix(M):
M:=normal(inverse(M)):
v:=convert([[1$2^(m-1)]],matrix):
vT:=convert([[1]$2^(m-1)],matrix):
normal(1+x*multiply(multiply(v,M),vT)[1,1]):

end:

#PaperRsv(M,x): an article with the generating functions for tiling an m by n rectangle by rectangles for m from 1 to M
#using the Joshua Smith-Helena Verrill approach. Try:
#PaperRsv(4,x);
PaperRsv:=proc(M,x) local m,gu,n,f,i:

print(`Generating functions for the number of ways to tile an m by n rectangle with Rectangular Tiles of any Size`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for m from 1 to M do
f:=FmxSV(m,x):

print(``):

print(`Theorem number`, m):
print(``):
print(`Let `, a[m](n), `be the number of ways to tile an`, m, `by n rectangle with RECTANGULAR tiles , then `):
print(``):
print(Sum(a[m](n)*x^n,n=0..infinity)=f):
print(``):
print(`and in Maple notation`):
print(``):
lprint(f):
print(``):

f:=taylor(f,x=0,33):
gu:=[seq(coeff(f,x,i),i=0..30)]:
print(`The first 31 terms of the sequence, starting at n=0 are `):
lprint(gu):

od:

print(``):
print(`----------------------`):
print(``):
print(`This ends this article that took`, time(), `seconds to generate `):
print(``):
end:

#End  the Smith-Verrill arroach


#DiGw(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the weight of each vertex.  Try:
#DiGw(2);
DiGw:=proc(m) local A,G,T,a,A2,Na,i,a1,g,T2:
A:=Alphabet(m):

G:=Grammar(m):

if A[1] minus A[2]<>{} then
 RETURN(FAIL):
fi:
if nops(A[3])<>1 then
 RETURN(FAIL):
fi:

a:=nops(A[2]):
A2:=convert(A[2],list):

Na[A[3][1]]:=a+2:
T2[1]:=0:

T2[a+2]:=nops(A[3][1]):

for i from 1 to a do
 Na[A2[i]]:=i+1:
  T2[i+1]:=nops(A2[i]):
od:

T[1]:={seq(Na[a1], a1 in A[1])}:

for i from 2 to a+1 do
 T[i]:={}:
od:

T[a+2]:={}:

for g in G[2] union G[3] do
 T[Na[g[1]] ]:=  T[Na[g[1]] ] union {Na[g[2]]}:
od:

[[seq(T[i],i=1..a+2)],[seq(T2[i],i=1..a+2)]]:

end:


#AdjMatW(WG,w): Given a weighted directed graph G=[G1,WeightTable], and a variable name w, (in our format n:=nops(G1)=nops(WeightTable). Outputs the n by n weighted adjacency matrix in terms of powers of w. Try
#Try:G1:=RDG(5,1/2);AdjMatW([G1,[0,2,3,4,2]]);

AdjMatW:=proc(G,w) local M,M1,i,n,j,G1,W1:
G1:=G[1]:
W1:=G[2]:

n:=nops(G1):

if nops(W1)<>n then
 RETURN(FAIL):
fi:

M:=[]:

for i from 1 to n do
M1:=[]:
for j from 1 to n do
 if member(j, G1[i]) then
   M1:=[op(M1),w^W1[j]]:
 else
   M1:=[op(M1),0]:
fi:
od:
M:=[op(M),M1]:
od:
M:
end:




#WalkGFw(G,x,w): Inputs a weighted directed graph G=[V,T], where T[v] is the set of outgoing neighbors of v, and a variable x, outputs the matrix of rational functions whose (i,j) entry is
#the generating function in x, according to lengths of walks from i to j. Try:
WalkGFw:=proc(G,x,w) local n,M,i:
n:=nops(G[1]):
M:=AdjMatW(G,w):
M:=matrix([seq(expand([0$(i-1),1,0$(n-i)]-x*M[i]),i=1..n)]):
normal(inverse(M))[1,n]:
end:


#Fmxw(m,x,w): The weighted generating function of SeqR(m,infinity); Try:
#Fmxw(3,x,w);
Fmxw:=proc(m,x,w) local G:
G:=DiGw(m):
normal(1+WalkGFw(G,x,w)/x):
end:



#CheckFmxw(m,N): checks the correctness of Fmxw(m,w,x) up to the coefficient of x^N. Try:
#CheckFmx2(2,5);
CheckFmxw:=proc(m,N) local f,x,w,n,gu1,gu2,S,s:
f:=Fmxw(m,x,w):
f:=taylor(f,x=0,N+1):
for n from 1 to N do
 gu1:=expand(coeff(f,x,n)):
 S:=TIrPe([n$m]):
 gu2:=add(w^nops(s), s in S):
 if gu1<>gu2 then
   print(`When n=`, n, ` it fails `):
   RETURN(false):
 fi:
od:
true:
end:

 


#FmxwPC(n,x,w): for n from 1 to 6 the pre-computed value of the bi-variate generating function SeqRgfW(n,x). Try:
#FmxwPC(5,x,w);
FmxwPC:=proc(n,x,w) local L:

if not n>=1 and n<=6 then
  RETURN(FAIL):
fi:
L:=
[-(w^4*x-w^3*x-w^2*x+1)/(w^3*x+w^2*x-1), -(2*w^10*x^2+2*w^9*x^2-2*w^8*x^2-2*w^7
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27*x^3-160566*w^26*x^4+96678*w^25*x^5-2558*w^24*x^6-2755*w^26*x^3-133950*w^25*x
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*w^22*x^6+5*w^25*x^2+7150*w^24*x^3-72753*w^23*x^4+11497*w^22*x^5-36*w^21*x^6+49
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1951*w^17*x^4+6*w^16*x^5-169*w^18*x^2+5660*w^17*x^3-799*w^16*x^4-507*w^17*x^2+
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6*w^14*x-626*w^13*x^2+224*w^12*x^3-4*w^13*x-452*w^12*x^2+90*w^11*x^3-w^12*x-287
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+25*w^9*x-41*w^8*x^2+29*w^8*x-18*w^7*x^2+21*w^7*x-6*w^6*x^2+14*w^6*x-2*w^5*x^2+
9*w^5*x+5*w^4*x+2*w^3*x+w^2*x-1)/(2*w^75*x^10+54*w^74*x^10+571*w^73*x^10+3581*w
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60*x^8-14595919*w^59*x^9+14371790*w^58*x^10+574923*w^59*x^8-20122804*w^58*x^9+
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58*x^7+3134290*w^57*x^8-30232853*w^56*x^9+9391014*w^55*x^10-1056*w^57*x^7+
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55*x^8-34102003*w^54*x^9+5031448*w^53*x^10-42501*w^55*x^7+15193317*w^54*x^8-\
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99*w^51*x^6-4640515*w^50*x^7+33802950*w^49*x^8-10195797*w^48*x^9+121480*w^47*x^
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364726734090*w^50*x^8-75625411391*w^49*x^9+2602566294*w^48*x^10-9110930*w^47*x^
11+328*w^46*x^12-194303*w^52*x^5+12304140792*w^51*x^6-216377093456*w^50*x^7+
231279240861*w^49*x^8-38021821034*w^48*x^9+996424963*w^47*x^10-2245398*w^46*x^
11+15*w^45*x^12-1161517*w^51*x^5+15154229539*w^50*x^6-172110476610*w^49*x^7+
141743414938*w^48*x^8-18419272214*w^47*x^9+361953097*w^46*x^10-494203*w^45*x^11
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^8-8584228531*w^46*x^9+124112049*w^45*x^10-94551*w^44*x^11-16834147*w^49*x^5+
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x^9+39915563*w^44*x^10-15078*w^43*x^11-44802769*w^48*x^5+18638833930*w^47*x^6-\
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*w^44*x^4-569745259*w^43*x^5+8631693295*w^42*x^6-6926773362*w^41*x^7+778479487*
w^40*x^8-11010355*w^39*x^9+5402*w^38*x^10+11171*w^43*x^4-678648619*w^42*x^5+
6432454135*w^41*x^6-3926677591*w^40*x^7+339604648*w^39*x^8-3327358*w^38*x^9+626
*w^37*x^10+72918*w^42*x^4-745045262*w^41*x^5+4594504241*w^40*x^6-2144036032*w^
39*x^7+141323436*w^38*x^8-921299*w^37*x^9+44*w^36*x^10+329458*w^41*x^4-\
760038137*w^40*x^5+3148997178*w^39*x^6-1126580575*w^38*x^7+55886694*w^37*x^8-\
229779*w^36*x^9+w^35*x^10+1091455*w^40*x^4-725195939*w^39*x^5+2072693566*w^38*x
^6-568962813*w^37*x^7+20900991*w^36*x^8-50338*w^35*x^9+2786178*w^39*x^4-\
650638112*w^38*x^5+1310861992*w^37*x^6-275738232*w^36*x^7+7347266*w^35*x^8-9306
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w^31*x^9+18650429*w^35*x^4-245533356*w^34*x^5+140680827*w^33*x^6-9569707*w^32*x
^7+49063*w^31*x^8-9*w^35*x^3+21670265*w^34*x^4-170313917*w^33*x^5+72656541*w^32
*x^6-3607151*w^31*x^7+10353*w^30*x^8-192*w^34*x^3+22920074*w^33*x^4-112875439*w
^32*x^5+35920694*w^31*x^6-1272444*w^30*x^7+1780*w^29*x^8-1826*w^33*x^3+22317876
*w^32*x^4-71535730*w^31*x^5+16962551*w^30*x^6-416077*w^29*x^7+223*w^28*x^8-\
10255*w^32*x^3+20181558*w^31*x^4-43370980*w^30*x^5+7628740*w^29*x^6-124481*w^28
*x^7+15*w^27*x^8-38126*w^31*x^3+17059776*w^30*x^4-25154805*w^29*x^5+3255190*w^
28*x^6-33433*w^27*x^7-100626*w^30*x^3+13547365*w^29*x^4-13949787*w^28*x^5+
1311260*w^27*x^6-7831*w^26*x^7-199999*w^29*x^3+10145626*w^28*x^4-7389289*w^27*x
^5+495375*w^26*x^6-1524*w^25*x^7-316612*w^28*x^3+7187292*w^27*x^4-3732758*w^26*
x^5+173957*w^25*x^6-225*w^24*x^7-419314*w^27*x^3+4827302*w^26*x^4-1793958*w^25*
x^5+56099*w^24*x^6-20*w^23*x^7-482138*w^26*x^3+3078345*w^25*x^4-817599*w^24*x^5
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459049*w^24*x^3+1073636*w^23*x^4-142046*w^22*x^5+889*w^21*x^6+6*w^24*x^2-391621
*w^23*x^3+586699*w^22*x^4-53383*w^21*x^5+146*w^20*x^6+94*w^23*x^2-309459*w^22*x
^3+303959*w^21*x^4-18446*w^20*x^5+15*w^19*x^6+584*w^22*x^2-228004*w^21*x^3+
148862*w^20*x^4-5758*w^19*x^5+1940*w^21*x^2-157122*w^20*x^3+68676*w^19*x^4-1580
*w^18*x^5+4062*w^20*x^2-101567*w^19*x^3+29679*w^18*x^4-362*w^17*x^5+6105*w^19*x
^2-61689*w^18*x^3+11921*w^17*x^4-64*w^16*x^5+7225*w^18*x^2-35209*w^17*x^3+4404*
w^16*x^4-6*w^15*x^5+7190*w^17*x^2-18859*w^16*x^3+1467*w^15*x^4+6355*w^16*x^2-\
9459*w^15*x^3+428*w^14*x^4+5066*w^15*x^2-4408*w^14*x^3+103*w^13*x^4+3683*w^14*x
^2-1903*w^13*x^3+19*w^12*x^4+2463*w^13*x^2-747*w^12*x^3+w^11*x^4-w^13*x+1511*w^
12*x^2-267*w^11*x^3-11*w^12*x+857*w^11*x^2-81*w^10*x^3-41*w^11*x+439*w^10*x^2-\
21*w^9*x^3-69*w^10*x+208*w^9*x^2-3*w^8*x^3-71*w^9*x+89*w^8*x^2-57*w^8*x+34*w^7*
x^2-38*w^7*x+10*w^6*x^2-27*w^6*x+3*w^5*x^2-14*w^5*x-7*w^4*x-3*w^3*x-w^2*x+1)]:

L[n]:

end:


#AsyAve(f,z,w,K): inputs a rational function f in z and w estiamtes [C1,C0] such that the coeff. of z^n of subs(w=1,diff(f,w)) divided by the coeff. of z^n  equals C1*n+C0 by looking
#at two consectutive terms around the coeff. of z^K. Try:
#AsyAve(1/(1-z*w-z^2),z,w,200);
AsyAve:=proc(f,z,w,K) local eq,var,C1,C0,f0,f1,f0T,f1T,C0a,C1a,C0b, C1b,sol,i:

f0:=normal(subs(w=1,f)):

f1:=normal(subs(w=1,diff(f,w))):
f0T:=taylor(f0,z=0,K+10):

f1T:=taylor(f1,z=0,K+10):

var:={C0,C1}:

eq:={seq(coeff(f1T,z,i)/coeff(f0T,z,i)=C0+C1*i,i=K-3..K-2)}:

sol:=solve(eq,var):

C0a:=subs(sol,C0):

C1a:=subs(sol,C1):



eq:={seq(coeff(f1T,z,i)/coeff(f0T,z,i)=C0+C1*i,i=K-1..K)}:

sol:=solve(eq,var):

C0b:=subs(sol,C0):

C1b:=subs(sol,C1):


if abs(C0a-C0b)>10^(-Digits-2) or abs(C1a-C1b)>10^(-Digits-2) then
  RETURN(FAIL):
fi:

evalf([C1b,C0b]):


end:


#HV(S): Given a set of edges of P_m x P_n outputs the pair [h,v] where h is the number of horizontal edges, and v is the number of vertical edges. Try:
#HV({{[1,0],[2,0]}}); HV({{[0,1],[0,2]}});

HV:=proc(S) local h,v,s:
h:=0:
v:=0:
for s in S do

if nops(s)<>2 then
 RETURN(FAIL):
fi:

if s[1][1]=s[2][1] then
 h:=h+1:
elif s[1][2]=s[2][2] then
 v:=v+1:
 else
 RETURN(FAIL):
fi:
od:
[h,v]:
end:





#DiGw1w2(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the bi-weight of each vertex ([number of horizontal edges in the grid, number of vertical edges in the grid].
#Try:
#DiGw1w2(2);
DiGw1w2:=proc(m) local A,G,T,a,A2,Na,i,a1,g,T2:
A:=Alphabet(m):

G:=Grammar(m):

if A[1] minus A[2]<>{} then
 RETURN(FAIL):
fi:
if nops(A[3])<>1 then
 RETURN(FAIL):
fi:

a:=nops(A[2]):
A2:=convert(A[2],list):

Na[A[3][1]]:=a+2:
T2[1]:=[0,0]:

T2[a+2]:=HV(A[3][1]):

for i from 1 to a do
 Na[A2[i]]:=i+1:
  T2[i+1]:=HV(A2[i]):
od:

T[1]:={seq(Na[a1], a1 in A[1])}:

for i from 2 to a+1 do
 T[i]:={}:
od:

T[a+2]:={}:

for g in G[2] union G[3] do
 T[Na[g[1]] ]:=  T[Na[g[1]] ] union {Na[g[2]]}:
od:

[[seq(T[i],i=1..a+2)],[seq(T2[i],i=1..a+2)]]:

end:



#AdjMatW1W2(G,w1,w2): Given a bi-weighted directed graph G=[G1,WeightBiTable], and a variable names w1, w2, (in our format n:=nops(G1)=nops(WeightTable). Outputs the n by n weighted adjacency matrix in terms of powers of w1 and w2. Try
# G:=DiGw1w2(2):AdjMatW1W2(G,w1,w2);

AdjMatW1W2:=proc(G,w1,w2) local M,M1,i,n,j,G1,W1:
G1:=G[1]:
W1:=G[2]:


n:=nops(G1):

if nops(W1)<>n then
 RETURN(FAIL):
fi:

M:=[]:

for i from 1 to n do
M1:=[]:
for j from 1 to n do
 if member(j, G1[i]) then
   M1:=[op(M1),w1^W1[j][1]*w2^W1[j][2]]:
 else
   M1:=[op(M1),0]:
fi:
od:
M:=[op(M),M1]:
od:
M:
end:




#WalkGFw1w2(G,x,w1,w2): Inputs a bi-weighted directed graph G=[V,T], where T[v] is the set of outgoing neighbors of v, and a variable x, outputs the matrix of rational functions whose (i,j) entry is
#the generating function in x, according to lengths of walks from i to j. Try:
WalkGFw1w2:=proc(G,x,w1,w2) local n,M,i:
n:=nops(G[1]):
M:=AdjMatW1W2(G,w1,w2):
M:=matrix([seq(expand([0$(i-1),1,0$(n-i)]-x*M[i]),i=1..n)]):
normal(inverse(M))[1,n]:
end:



#Fmxw1w2(m,x,w1,w2): The bi-weighted generating function (by w1^NumberOfHorizontalEdges*w2^NumberIfVerticalDdeges  of SeqR(m,infinity); Try:
#Fmxw1w2(3,x,w1,w2);
Fmxw1w2:=proc(m,x,w1,w2) local G:
G:=DiGw1w2(m):
normal(1+WalkGFw1w2(G,x,w1,w2)/x):
end:



#CheckFmxw1w2(m,N): checks the correctness of Fmxw1w2(m,w1,w2,x) up to the coefficient of x^N. Try:
#CheckFmxw1w2(2,5);
CheckFmxw1w2:=proc(m,N) local f,x,w1,w2,n,gu1,gu2,S,s:
f:=Fmxw1w2(m,x,w1,w2):
f:=taylor(f,x=0,N+1):
for n from 1 to N do
 gu1:=expand(coeff(f,x,n)):
 S:=TIrPe([n$m]):
 gu2:=add(w1^HV(s)[1]*w2^HV(s)[2], s in S):
 if gu1<>gu2 then
   print(`When n=`, n, ` it fails `):
   RETURN(false):
 fi:
od:
true:
end:


#NewTileSi(i,k): the signal of a new tile in a letter starting with horizontal edge {[i,0],[i,1]} and ending in horizontal edge {[i+k,0],[i+k,1]}, it outputs the
#set of edges that should be there followed by the set of horizontal edges that should not. Try:
#NewTileSi(2,4);
NewTileSi:=proc(i,k) local j:
[{seq({[i+j,0],[i+j+1,0]},j=0..k-1),{[i,0],[i,1]},{[i+k,0],[i+k,1]}},{seq({[i+j,0],[i+j,1]},j=1..k-1)}]:
end:


#NNT(LE): The number of new tiles starting with the letter LE in Alphabet(m)
#Try:
#LE:=Alphbaet(4)[2][1]; NT(LE);
NNT:=proc(m,LE) local i,co,k,gu:
co:=0:

for i from 0 to m-1 do
for k from 1 to m-i do
 gu:=NewTileSi(i,k):
 if gu[1] minus LE={} and gu[2] intersect LE={} then
  co:=co+1:
 fi:
od:
od:
co:

end:





####


#DiGt(m): the directed graph representation of the grammar of tilings of [m]x[n] by rectangles for a fixed m, followed by the "numtber of new tiles" weight of each vertex.  Try:
#DiGt(2);
DiGt:=proc(m) local A,G,T,a,A2,Na,i,a1,g,T2:
A:=Alphabet(m):

G:=Grammar(m):

if A[1] minus A[2]<>{} then
 RETURN(FAIL):
fi:
if nops(A[3])<>1 then
 RETURN(FAIL):
fi:

a:=nops(A[2]):
A2:=convert(A[2],list):

Na[A[3][1]]:=a+2:
T2[1]:=0:

T2[a+2]:=NNT(m,A[3][1]):

for i from 1 to a do
 Na[A2[i]]:=i+1:
  T2[i+1]:=NNT(m,A2[i]):
od:

T[1]:={seq(Na[a1], a1 in A[1])}:

for i from 2 to a+1 do
 T[i]:={}:
od:

T[a+2]:={}:

for g in G[2] union G[3] do
 T[Na[g[1]] ]:=  T[Na[g[1]] ] union {Na[g[2]]}:
od:

[[seq(T[i],i=1..a+2)],[seq(T2[i],i=1..a+2)]]:

end:


#Fmxt(m,x,w): The weighted generating function, according to the weight t^NumberOfTIles of SeqR(m,infinity); Try:
#Fmxt(3,x,t);
Fmxt:=proc(m,x,t) local G:
G:=DiGt(m):
normal(1+WalkGFw(G,x,t)/x):
end:



#CheckFmxt(m,N): checks the correctness of Fmxt(m,w,t) up to the coefficient of x^N. Try:
#CheckFmx2(2,5);
CheckFmxt:=proc(m,N) local f,x,t,n,gu1,gu2,S,s:
f:=Fmxt(m,x,t):
f:=taylor(f,x=0,N+1):
for n from 1 to N do
 gu1:=expand(coeff(f,x,n)):
 S:=TIrP([n$m]):
 gu2:=add(t^nops(s), s in S):
 if gu1<>gu2 then
   print(`When n=`, n, ` it fails `):
   RETURN(false):
 fi:
od:
true:
end: