##############################################################################
# GenBeukersZeta2.txt Save this file as  GenBeukersZeta2.txt  to use it,      #
# stay in the                                                                 #
## same directory, get into Maple (by typing: maple <Enter> )                 #
## and then type:  read GenBeukersZeta2.txt <Enter>                           #
## Then follow the instructions given there                                   #
##                                                                            #
## Written by Robert Dougherty-Bliss and Doron Zeilberger                     #
#Rutgers University ,                                                         #                                          
#and Christoph Koutshan, Linz                                                 #
#robert dot w dot bliss at gmail dot com                                      #
##  DoronZeil at gmail dot com                                                # 
###############################################################################


print(` Written: Sept./Dec 2020 `):
print():
print(`This is GenBeukersZeta2.txt, A Maple package for efficient computation of generalized Beukers integrals`):
print(`that generalize the Beukers Zeta(2) integral `):

print(` It experiments with the Generalized Beukers integral`):
print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(``):
print(`It also considers the even more general integral`):
print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(``):
print(`where the case c=0 correspond to the former case`):

print(` It is one of the Maple packages that accompany   the article `):
print(`Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs A La Apery`):
print(``):
print(`avaliable from the authors' websites and from arxiv.org `):
print(``):

print():
print(`The most current version is available on WWW at:`):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/GenBeukers.txt .`):
print(`Please report all bugs to: DoronZeil at gmail dot com .`):



print():
print(`---------------------------------------------`):
print():
print(`For a list of the STORY  procedures`):
print(` type "ezraST();". For specific help type "ezra(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():



print():
print(`---------------------------------------------`):
print():
print(`For a list of the PRIMES  procedures`):
print(` type "ezraPR();". For specific help type "ezra(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():


print():
print(`---------------------------------------------`):
print():
print(`For a list of the Pre-Computed  procedures`):
print(` type "ezraPC();". For specific help type "ezra(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():


print():
print(`---------------------------------------------`):
print():
print(`For a list of the  procedures from MultiAlmkvistZeilbeger.txt `):
print(` type "ezraMAZ();". For specific help type "ezraMAZ(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():


print():
print(`---------------------------------------------`):
print():
print(`For a list of the Generalized  procedures`):
print(` type "ezraG();". For specific help type "ezra(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():

print():
print(`---------------------------------------------`):
print():
print(`For a list of the SUPPORTING procedures`):
print(` type "ezra1();". For specific help type "ezra(procedure_name);" `):
print():
print():
print(`---------------------------------------------`):
print():
 

print():
print(`---------------------------------------------`):
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():
print():
print(`---------------------------------------------`):
print():
 



ezraPR:=proc()
if args=NULL then
 print(` The PRIMES pocedures are`):
 print(` AsyPpG, EvalPpG, PpG, PpGlimit, PrimesF, PrintPpG `):
else
 ezra(args):
fi:

end:


ezraST:=proc()
if args=NULL then
 print(` The Story procedures are`):
 print(`Info1, Mamar2, Mamar3, Mamar4, PaperZ2, PaperZ2g, PrintPpG, TheoremZ2, TheoremZ2g, TheoremZ2gnum, TheoremZ2num,  WadimHWv,  WadimHWvLP, `):
else
 ezra(args):
fi:

end:

ezraG:=proc()
if args=NULL then
 print(` generalized procedures are:`):
 print(` AnBnG, AppxSeqG, CnDnG, deltSeqG, FindRelG,  hopefulSearchG, IntGBg, GBCg, OPEZ2g `):
else
 ezra(args):
fi:

end:


ezraPC:=proc()
if args=NULL then
 print(`The pre-computed procedures are:`):
 print(` AlmostHopefuls1G, Hopefuls1, Hopefuls1C, Hopefuls1G, Hopefuls2, Hopefuls2C, Hopefuls3, Hopefuls3C, Hopefuls4, Hopefuls4C, Hopefuls5, Hopefuls5C `):
 print(`Hopefuls5v, WadimHW `):
else
 ezra(args):
fi:

end:

ezra1:=proc()
if args=NULL then
 print(`The supporting procedures are:`):
 print(` AnBnSym,AppxSeq1, FindRel, FindRelSym, FindRelStupid `):
 print(`FracPts, GBC1, GuessK1, `):
 print(` hopefulSearch,  hopefulSearchP, HopeStatus, IntGBnaked, IntGBser,IntGBsummand, IntGBsummandRF, IntGB3F2 `):
 print(`IntGBser, Khez, LeadA, LC, NormalizePair, Nu1,  rf, SortC  `):
else
 ezra(args):
fi:

end:

ezra:=proc()
if args=NULL then

print(`The MAIN procedures are:`) :
print(` AnBn, AppxSeq, BC, CnDn, delt, deltSeq, GBC, IntGB, MyIDs, OPEZ2,  Search, SeqFromRec  `):





elif nargs=1 and args[1]=AlmostHopefuls1G then
print(`AlmostHopefuls1G(): A list of 3  pentuples with negative but small deltas for the generalized case. Try:`):
print(`AlmostHopefuls1G();`):


elif nargs=1 and args[1]=AnBn then
print(`AnBn(a1,a2,b1,b2,K): The first K terms in the sequences An and Bn in the expression`):
print(`IntGB(a1,a2,b1,b2,n)=B(n)*c-A(n)`):
print(`where c=IntGB(a1,a2,b1,b2,0);`):
print(`It also returns the first K terms in the floating point of the sequence IntGB(a1,a2,b1,b2,n) itself. Try: `):
print(`AnBn(0,0,0,0,50);`):
print(`AnBn(1/2,0,0,1/2,50);`):
print(`AnBn(2/3,1/3,1/6,1/3,50);`):


elif nargs=1 and args[1]=AnBnG then
print(`AnBnG(a1,a2,b1,b2,c,K): The first K terms in the sequences An and Bn in the expression`):
print(`IntGBg(a1,a2,b1,b2,n)=B(n)*CONST-A(n)`):
print(`where CONT=IntGBg(a1,a2,b1,b2,c,0);`):
print(`It also returns the first K terms in the floating point of the sequence IntGB(a1,a2,b1,b2,n) itself. Try: `):
print(`AnBnG(0,0,0,0,1,50);`):
print(`AnBnG(0,0,0,0,1/2,50);`):


elif nargs=1 and args[1]=AnBnSym then
print(`AnBnSym(a1,a2,b1,b2,K): The first K terms in the sequences An and Bn in the expression`):
print(`IntGB(a1,a2,b1,b2,n)=B(n)*c-A(n)`):
print(`where c=IntGB(a1,a2,b1,b2,0);`):
print(`FOR SYMBOLIC a1,a2,b1,b2. Try:`):
print(`AnBnSym(a1,a2,b1,b2,50);`):

elif nargs=1 and args[1]=AppxSeq then
print(`AppxSeq(a1,a2,b1,b2,K): The approximating sequence for  IntGB(a1,a2,b1,b2,0); `):
print(`Try: `):
print(` AppxSeq(0,0,0,0,40); `):
print(` AppxSeq(1/2,0,0,1/2,40); `):
print(` AppxSeq(2/3,1/3,1/6,1/3,40); `):


elif nargs=1 and args[1]=AppxSeqG then
print(`AppxSeqG(a1,a2,b1,b2,c,K): The approximating sequence for  IntGBg(a1,a2,b1,b2,c,0); `):
print(`Try: `):
print(` AppxSeqG(0,0,0,0,0,40); `):
print(` AppxSeqG(1/2,0,0,1/2,0,40); `):
print(` AppxSeqG(2/3,1/3,1/6,1/3,0,40); `):
print(` AppxSeqG(2/3,1/3,1/6,1/3,1/2,40); `):

elif nargs=1 and args[1]=AppxSeq1 then
print(`AppxSeq1(ope,n,N,RE,K): It inputs a second order operator ope, in n where N is the shift operator N`):
print(`where it is known that ope(n,N) annihilates a sequence a(n) that goes to zero.`):
print(`It also inputs a pair RE=[[c0,c1],L] satisfying c0*a(0)+c1*a(1)=L where c0,c1 are integers and`):
print(`L is a rational number, It outputs the list of the first K terms in the rational approximations to a(0). Try`):
print(`AppxSeq1(N^2-N-1,n,N,[[11,12],3],20);`):


elif nargs=1 and args[1]=AsyPpG then
print(`AsyPpG(P): inputs a  GpP expression P=[PA,PB] and outputs the`):
print(`EXACT value of limit of log( EvalPpG(P,n))/n as n goes to infinity. Try`):

print(`The following examples are parts of irrationality proofs, i.e. yielding positive delta, albeit of famous constants`):
print(``):
print(`For GBC(0,0,1/2,0)  (alias 2*log(2)) , try: `):
print(`AsyPpG([ [[2,2],[1,0],[]],[[1, 1], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ]]);`):

print(``):
print(`For GBC(0,0,1/3,-2/3) try: `):
print(`AsyPpG([[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {2}, 3]]],[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$2, [0, 1/3, 0, 3/2, {0}, 1], [2/3, 1, 0, 3/2, {0}, 1]]]]);`);

print(`For the equivalent GBC(0,-2/3,2/3,-2/3), but much simpler, try:`):
print(`AsyPpG([[[1, 0], [1, 0], []],[[3, 3/2], [1, 0], [[0, 1/3, 0, 1, {0}, 1]]]]);`);

print(``):
print(`For GBC(-3/4,-3/4,-1/4,-1/4),  conjecturally 92-64*sqrt(2), try: `):
print(` AsyPpG([[[1, 0], [1, 0], []], [[2, 2], [1, 0], [[1/2, 3/4, 0, 2, {0}, 1]] ]]); `):


#print(`For GBC(0,-1/2,1/6,-1/2), type:`): #SOMETHING IS WRONG, TO BE CHECKED
#print(`AsyPpG([[[3, 3/2], [1, 0], []],[[2, 2], [1, 0], [[0, 1/3, 0, 1, {0}, 1]]]]);`);


print(`------------------------------`):

print(`The following examples give negative deltas, but they are still interesting`):

print(`For GBC(1/2,-1/2,-1/2,1/2), (conjecturally 4-8/Pi) , type:`):
print(`AsyPpG([[[1, 0], [1, 0], []],[[2, 4], [1, 0], [[0, 1/2, 0, 1, {0}, 1]]]]);`);


print(`For GBC(1/2,0,0,1/2), twice the Catalan constant , type:`):
print(`AsyPpG([[[1, 0], [1, 0], []],[[2, 4], [2, 2], [ ]]]);`);


print(`For GBC(1/3,0,0,1/3), whatever it is , type:`):
print(`AsyPpG([[[1, 0], [1, 0], []],[[3,3], [3/2, 2], [  [1/3,2/3,3/2,3,{2},3]$2     ]    ]]);`);


elif nargs=1 and args[1]=AsyPpG1 then
print(`AsyPpG1(P): inputs a partial GpP expression  and outputs the`):
print(`EXACT value of limit of log( EvalPpG1(P,n))/n as n goes to infinity. Try`):
print(`For GBC(0,0,1/2,0) try: `):
print(`AsyPpG1([[2, 2], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ] );`):

elif nargs=1 and args[1]=BC then
print(`BC(): The Beukers constant 11/2+5*sqrt(5)/2`):

elif nargs=1 and args[1]=CnDn then
print(`CnDn(a1,a2,b1,b2,K): The first K terms in the sequences gcd(numer(An[i]),numer(Bn[i]))`):
print(`lcm(denom(An[i]),denom(Bn[i]), followed by the asympototics of the normalizing sequence, and the implied deltas.). Try:`):
print(` that approximate IntGB(a1,a2,b1,b2,0);`):
print(`The output is a list of length 3 where the first one is the pair of sequences `):
print(`[lcm(numer(An[i]),numer(Bn[i])) `):
print(``):
print(`The second entry is the last 20 terms of  the natural log of the  normalizing sequence`):
print(``):
print(`The third entryis the last 20 terms of  the conservative delta. If they are positive it indicates that there is probably`):
print(`an inrrationality proof, modulo a certain divisibility lemma that needs a human, like Wadim Zudilin, to prove.`):
print(``):
print(`CnDn(0,0,0,0,50); `):
print(` CnDn(1/2,0,0,1/2,50); `):
print(` CnDn(1/2,0,0,1/2,50); `):
print(`CnDn(2/3,1/3,1/6,1/3,50);`):


elif nargs=1 and args[1]=CnDnG then
print(`CnDnG(a1,a2,b1,b2,c,K): The first K terms in the sequences gcd(numer(An[i]),numer(Bn[i]))`):
print(`lcm(denom(An[i]),denom(Bn[i]), followed by the asympototics of the normalizing sequence, and the implied deltas.). Try:`):
print(` that approximate IntGBg(a1,a2,b1,b2,c);`):
print(`The output is a list of length 3 where the first one is the pair of sequences `):
print(`[lcm(numer(An[i]),numer(Bn[i])) `):
print(`CnDnG(0,0,0,0,0,50); `):
print(` CnDnG(1/2,0,0,1/2,0,50); `):
print(` CnDnG(1/2,0,0,1/2,0,50); `):
print(`CnDnG(2/3,1/3,1/6,1/3,0,50);`):


elif nargs=1 and args[1]=CnDnG then
print(`CnDnG(a1,a2,b1,b2,c,K): The first K terms in the sequences gcd(numer(An[i]),numer(Bn[i]))`):
print(`lcm(denom(An[i]),denom(Bn[i])). that approximate IntGBg(a1,a2,b1,b2,c,0); Try:`):
print(`It also returns the last 20 terms of the logs of the normailizing sequence`):
print(`CnDnG(0,0,0,0,0,50); `):
print(`CnDnG(0,0,0,0,1/2,50); `):


elif nargs=1 and args[1]=delt then
print(` delt(a,c): Given a rational number a and a constant `):
print(` c, finds the delta such that |a-c|=1/denom(a)^(1+delta) `):
print(` For example, try delt(22/7,evalf(Pi)); `):



elif nargs=1 and args[1]=deltSeq then
print(`deltSeq(a1,a2,b1,b2,K): The sequence for the  empirical deltas for  IntGB(a1,a2,b1,b2,0); `):
print(`Try: `):
print(`The folllowing is good old Apery-Beukers for Zeta(2) `):
print(` deltSeq(0,0,0,0,40); `):
print(`This one is Zudilin's construction for Catalan's constant`):
print(` deltSeq(1/2,0,0,1/2,40); `):
print(`This would give yet another proof for the irrationality of log(2) [not as good at Alladi]`):
print(` deltSeq(-1/2,1/2,0,-1/2,50); `):
print(`This seems to indicate irrationality proof for IntGB(2/3,1/3,1/6,1/3,0)`):
print(` deltSeq(2/3,1/3,1/6,1/3,50); `):
print(`This seems to indicate irrationality proof for IntGB(-1/3,-1/2,-2/3,-1/2,0)`):
print(` deltSeq(-1/3,-1/2,-2/3,-1/2,50); `):


elif nargs=1 and args[1]=deltSeqG then
print(`deltSeqG(a1,a2,b1,b2,K): The sequence for the  empirical deltas for  IntGBg(a1,a2,b1,b2,c,0); `):
print(`Try: `):
print(`The folllowing is good old Apery-Beukers for Zeta(2) `):
print(` deltSeqG(0,0,0,0,0,40); `):
print(`This one is Zudilin's construction for Catalan's constant`):
print(` deltSeqG(1/2,0,0,1/2,0,40); `):
print(`For a case when  c is not 0, try:`):
print(` deltSeqG(0,0,0,0,1/2,40); `):


elif nargs=1 and args[1]=EvalPpG1 then
print(`EvalPpG1(P,n1): evaluate a partial (either top or bottom) PpG object at n=n1. `):
print(`For GBC(0,0,0,0) (alias Zeta(2))`):
print(`EvalPpG1([[1, 1], [1, 2],[] ],1000  );`):
print(``):
print(`For GBC(0,0,1/2,0) try: `):
print(`EvalPpG1([[1/2, 2], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ],1000 );`):



elif nargs=1 and args[1]=EvalPpG then
print(`EvalPpG(P,n1): evaluate a PpG  PA=[P1,P2], where P1 and P2 are the partial expressions for Cn and Dn resp.,  at n=n1. `):
print(`For GBC(0,0,0,0) (alias Zeta(2))`):
print(`EvalPpG([ [[1,1],[1,0],[]],    [[1, 1], [1, 2],[] ]],1000  );`):
print(``):
print(`For GBC(0,0,1/2,0) (alias 2*log(2)), try: `):
print(`EvalPpG([ [[2,2],[1,0],[]],[[1, 1], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ]],1000 );`):
print(``):
print(`For GBC(0,0,1/3,-2/3) try: `):
print(``):
print(`For GBC(0,0,1/3,-2/3) try: `):
print(`EvalPpG([[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {2}, 3]]],[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$2, [0, 1/3, 0, 3/2, {0}, 1], [2/3, 1, 0, 3/2, {0}, 1]]]],2000);`);

print(`For the equivalent GBC(0,-2/3,2/3,-2/3), but much simpler, try:`):
print(`EvalPpG([[[1, 0], [1, 0], []],[[3, 3/2], [1, 0], [[0, 1/3, 0, 1, {0}, 1]]]],2000);`);

print(``):
print(`For GBC(-3/4,-3/4,-1/4,-1/4),  conjecturally 92-64*sqrt(2), try: `):
print(` EvalPpG([[[1, 0], [1, 0], []], [[2, 2], [1, 0], [[1/2, 3/4, 0, 2, {0}, 1]] ]],2000); `):



elif nargs=1 and args[1]=FindRelStupid then
print(`FindRelStupid(a1,a2,b1,b2,K): Tries to find a pair of integers [c0,c1] where 1<=c0<=K, -K<=c1<=K such that`):
print(`c0*IntGB(a1,a2,b1,b2,0)+c1*IntGB(a1,a2,b1,b2,1) is a rational number. `):
print(`It returns r [[c0,c1],IntegralWithThatChoice]. Try:`):
print(`FindRelStupid(0,0,0,0,3);`):
print(`FindRelStupid(1/2,0,0,1/2,7);`):
print(`FindRelStupid(1/3,0,1/3,0,7);`):
print(`FindRelStupid(1/4,0,1/4,0,5);`):


elif nargs=1 and args[1]=FindRel then
print(`FindReCl(a1,a2,b1,b2): Tries to find a pair of integers [c0,c1] `):
print(`c0*IntGB(a1,a2,b1,b2,0)+c1*IntGB(a1,a2,b1,b2,1) is a rational number. `):
print(`It returns r [[c0,c1],IntegralWithThatChoice]. Try:`):
print(`FindRel(0,0,0,0);`):
print(`FindRel(1/2,0,0,1/2);`):
print(`FindRel(1/3,0,1/3,0);`):
print(`FindRel(1/4,0,1/4,0);`):
print(`FindRel(2/3,1/3,1/6,1/3);`):


elif nargs=1 and args[1]=FindRelG then
print(`FindRelG(a1,a2,b1,b2,c): A clever way to find a relationship`):
print(`between IntGBg(a1,a2,b1,b2,c,0) and  IntGBg(a1,a2,b1,b2,c,1) . Try:`):
print(`FindRelG(1/2,0,0,1/2,0);`):

elif nargs=1 and args[1]=FindRelSym then
print(`FindRelSym(a1,a2,b1,b2): A clever way to find a relationship`):
print(`between IntGB(a1,a2,b1,b2,0) and  IntGB(a1,a2,b1,b2,1)  FOR SYMBOLIC a1,a2,b1,b2. Try:`):
print(`FindRelSym(a1,a2,b1,b2);`):


elif nargs=1 and args[1]=FracPts then
print(`FracPts(n,S,K): Given a large positive integer n, a set of primes S and a pos. const. K`):
print(`outputs the set frac(n/p) (in floating point) for all p in S larger than sqrt(K*n). Try:`):
print(`FracPts(ithprime(1000),{seq(ithprime(i),i=1..1000)},7);`):

elif nargs=1 and args[1]=GuessK then
print(`GuessK(L): Given a list of rational numbers finds a rational number  such that L[i]*K^i has no factors that are`):
print(` large powers in the denominator or numerator`):
print(`Try: `):
print(`GuessK([seq(7*5^i/3^i,i=1..50)]); `):

elif nargs=1 and args[1]=GuessK1 then
print(`GuessK1(L): Given a list of integers finds an integer K such that L[i]/K^i is still an integer or fixed denominator`):
print(`Try: `):
print(`GuessK1([seq(7*5^i,i=1..50)]); `):


elif nargs=1 and args[1]=Hopefuls1 then
print(`Hopefuls1(): A list of 167 promising quadruples with positive empirical delta. Try:`):
print(`Hopefuls1();`):

elif nargs=1 and args[1]=Hopefuls1C then
print(`Hopefuls1C(r): A list of length 21 of equivalence classes of qudruples based on Hopefuls1(). Try:`):
print(`Hopefuls1C(r);`):


elif nargs=1 and args[1]=Hopefuls1G then
print(`Hopefuls1G(): A list of 4 promising pentuples with positive empirical delta for the generalized case. Try:`):
print(`Hopefuls1G();`):


elif nargs=1 and args[1]=Hopefuls2 then
print(`Hopefuls2(): A list of 123 VERY promising quadruples with positive stricted empirical delta. Try:`):
print(`Hopefuls2();`):


elif nargs=1 and args[1]=Hopefuls2C then
print(`Hopefuls2C(r): A list of length 16 of equivalence classes of qudruples based on Hopefuls2(). Try:`):
print(`Hopefuls2C(r);`):

elif nargs=1 and args[1]=Hopefuls3 then
print(`Hopefuls3(): A lis of 35 tuples [a1,a2,b1,b2] for which there is a potential irrationality proof `):
print(`for GBC(a1,a2,b1,b2) (q.v.) AND Maple can NOT identitify it  in terms of algebraic and/or combinations of log `):
print(`[Of course, just because Maple can't do it does not mean that it can't be done.] `):
print(``):
print(`hence there is A HOPE for a NEW irrationality theorem for a natural  constant defined by a double integral (possibly divided by Gammas).Try:`):
print(`Hopefuls3();`):


elif nargs=1 and args[1]=Hopefuls3C then
print(`Hopefuls3C(r): A list of length 9 of equivalence classes of qudruples based on Hopefuls3(). Try:`):
print(`Hopefuls3C(r);`):

elif nargs=1 and args[1]=Hopefuls4 then
print(`Hopefuls4(): A list of 88 tuples [a1,a2,b1,b2] for which there is a potential irrationality proof `):
print(`for GBC(a1,a2,b1,b2) (q.v.) and Maple CAN identitify in terms of algebraic and/or combinations of log `):
print(`hence form candiates for a NEW irrationality theorem for a natural  constant.Try:`):
print(`Hopefuls4();`):


elif nargs=1 and args[1]=Hopefuls4C then
print(`Hopefuls4C(r): A list of length 8 of equivalence classes of qudruples based on Hopefuls4(). Try:`):
print(`Hopefuls4C(r);`):


elif nargs=1 and args[1]=Hopefuls5 then
print(`Hopefuls5(): A list of lists of hopefuls parameters of length 6 whose i-th entry  have denominators i+1. Try: `):
print(`Hopefuls5();`):


elif nargs=1 and args[1]=Hopefuls5C then
print(`Hopefuls5C(r): A list of length 6 based on Hopefuls5(). Try:`):
print(`Hopefuls5C(r);`):

elif nargs=1 and args[1]=Hopefuls5v then
print(`Hopefuls5v(K): Some constants that probably have an irrationality proof, indicating those that are not yet identified.`):
print(`together for each the relevant information. K should be at least 1000 (it is the parameter used to collect data). Try:`):
print(`Hopefuls5v(2000):`):

elif nargs=1 and args[1]=hopefulSearch then
print(`hopefulSearch(numers, denoms, eps=0.001, blockPosInts=true, report=true):`):
print(`Exhaustively searches a collection of rational parameters to IntGB for`):
print(`promising irrationality measures. Searches all p / q where p is in`):
print(`'numers' and q is in 'denoms' and reports deltas > 'eps'. Optionally blocks`):
print(`positive integers (which often results in division by zero errors).`):
print(`Try:`):
print(`hopefulSearch([0, 1], [2, 3]);`):
print(`hopefulSearch([0, 1], [2, 3], eps=0.1); `):
print(`hopefulSearch([0, 1, 2, 3], [2, 3, 4, 5]); `):

elif nargs=1 and args[1]=hopefulSearchG then
print(`hopefulSearchG(numers, denoms, c_numers, c_denoms, eps=0.001, blockPosInts=true, report=true):`):
print(`Exhaustively searches a collection of rational parameters to IntGBg for`):
print(`promising irrationality measures. For the first (not c) parameters,`):
print(`searches all p / q where p is in 'numers' and q is in 'denoms' and reports deltas > 'eps'.`):
print(`For c, searches p / q where p is in 'c_numers' and q is in 'c_denoms'.`):
print(`Skips whenever p / q is a positive integer (which often results in division by zero errors).`):
print(`Try:`):
print(`hopefulSearchG([0, 1], [2, 3], [1, 2], [1, 2]);`):
print(`hopefulSearchG([0, 1], [2, 3], [1, 2], [1, 2], eps=0.1);`):
print(`hopefulSearchG([0, 1], [2, 3], [1, 2, 3], [1, 2, 3]);`):


elif nargs=1 and args[1]=hopefulSearchP then
print(`hopefulSearchP(numers, denoms):`):
print(`Like hopefulSearch(numers, denoms) (q.v.) but for each hopeful quartet, it does not only give the empirical delta`):
print(`but also the identified (or not) value, by Maple (so only conjectured)  of the constant in-question, and what Maple thinks `):
print(`is the exact value, rigorously, but often it gives you garbage`):
print(`Try:`):
print(`hopefulSearchP([0, 1], [2, 3]);`):
print(`hopefulSearchP([0, 1, 2, 3], [2, 3, 4, 5]); `):


elif nargs=1 and args[1]=HopeStatus then
print(`HopeStatus(a1,a2,b1,b2,K): inputs a1,a2,b1,b2, such that [a1,a2,b1,b2] is hopeful, checks the`):
print(`further rating for the more conserative test of CnDn(a1,a2,b1,b2,K)[3]. If they`):
print(`all negative then it returns 0 followed by the smallest neative. If some are negative and some positive, it returns the`):
print(`1/2 followed by the smallest negative and larger positive, if they are all positive`):
print(`it returns 1 followed by the smallest positive. Try:`):
print(`HopeStatus(5/3,5/3,0,0,2000);`):


elif nargs=1 and args[1]=Info1 then
print(`Info1(a1,a2,b1,b2,K): inputs a four-tuple (a1,a2,b1,b2) and outputs a list consisting of`):
print(`(i) the constant as a 3F2, (ii) the floating point to 100 digits,`):
print(`(iii) the recurrence operator annihilating An and Bn (iv) the pair of initial conditions for An and Bn`):
print(`i.e. [[A(0),A(1)],[B(0),B(1)]] (v) the estimated value of the limit of the log of the INTEGERATING functor divided by n (vi) the`):
print(`estimated implied irrationality measure using K values (vii) an identification if successful. Try:`):
print(`Info1(0,0,0,0,2000);`):

elif nargs=1 and args[1]=IntGB then
print(`IntGB(a1,a2,b1,b2,n1); `):
print(`int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1);`):
print(`divided by Beta(1-a1,1-a2)*Beta(1-b1,1-b2) `):
print(`Try: `):
print(` IntGB(0,0,0,0,0); `):


elif nargs=1 and args[1]=IntGBg then
print(`IntGBg(a1,a2,b1,b2,c,n1);`):
print(`int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1) divided by`):
print(`int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^c,x=0..1),y=0..1) `):
print(`Try: `):
print(`IntGBg(0,0,0,0,0,0);`):

elif nargs=1 and args[1]=IntGBnaked then
print(`IntGBnaked(a1,a2,b1,b2,n1); `):
print(`int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1); Try:`):
print(` IntGBnaked(0,0,0,0,0); `):

elif nargs=1 and args[1]=IntGBsummand then
print(`IntGBsummand(a1,a2,b1,b2,n): The summand in the series representation for IntGB(a1,a2,b1,b2,0); Try:`):
print(`IntGBsummand(0,0,0,0,n);`):
print(`IntGBsummand(1/2,0,0,1/2,n);`):


elif nargs=1 and args[1]=IntGB3F2 then
print(`IntGB3F2(a1,a2,b1,b2): Expressing GBC(a1,a2,b1,b2) as a 3F2, the output is a pair of lists, the first of size 3, the second of size 2`):
print(`corresponding to the numerator and denominator parameters, respectively. Try:`):
print(`IntGB3F2(0,0,0,0);`):
print(`IntGB3F2(1/2,0,0,1/2);`):


elif nargs=1 and args[1]=IntGBsummandRF then
print(`IntGBsummandRF(a1,a2,b1,b2,n,RF): The summand in the series representation for IntGB(a1,a2,b1,b2,0) in terms of the raising-factorial`):
print(`denoted by RF. Try: `):
print(`IntGBsummandRF(0,0,0,0,n,RF);`):
print(`IntGBsummandRF(1/2,0,0,1/2,n,RF);`):



elif nargs=1 and args[1]=GBC then
print(`GBC(a1,a2,b1,b2): A floating-point approximation to the constant`):
print(`IntGB(a1,a2,b1,b2,0). Try: `):
print(`GBC(0,0,0,0);`):
print(` GBC(1/2,0,0,1/2); `):
print(`This gives a multiple of 3/2*2^(1/3): `):
print(`GBC(2/3,1/3,1/6,1/3);`):


elif nargs=1 and args[1]=GBCg then
print(`GBCg(a1,a2,b1,b2,c): A floating-point approximation to the constant`):
print(`IntGBg(a1,a2,b1,b2,c,0). Try: `):
print(`GBCg(0,0,0,0,0);`):
print(` GBCg(1/2,0,0,1/2,0); `):
print(`This gives a multiple of 3/2*2^(1/3): `):
print(`GBCg(2/3,1/3,1/6,1/3,0);`):



elif nargs=1 and args[1]=IntGBser then
print(`IntGBser(a1,a2,b1,b2,K): an approximation to GBC(a1,a2,b1,b2) using the series with K terms. Try:`):
print(`IntGBser(0,0,0,0,100);`):
print(`IntGBser(1/2,0,0,1/2,100);`):

elif nargs=1 and args[1]=Khez then
print(`Khez(n,p): The largest exponent i such that n/p^i is an integer. Try:`):
print(`Khez(100,5);`):

elif nargs=1 and args[1]=LeadA then
print(` LeadA(a1,a2,b1,b2,n): The leading asymptotics in n in the terms of the series IntGBser(a1,a2,b1,b2). `):
print(` LeadA(0,0,0,0,n);`):

elif nargs=1 and args[1]=LC then
print(`LC(n): The lcm of 1...n `):

elif nargs=1 and args[1]=Mamar2 then
print(`Mamar2(K) : PaperZ2(L,K): where L is given by Hopefuls2(): Try:`):
print(`Mamar2(2000):`):


elif nargs=1 and args[1]=Mamar2 then
print(`Mamar2: PaperZ2(L,K): where L is given by Hopefuls2(): Try:`):
print(`Mamar2(2000):`):


elif nargs=1 and args[1]=Mamar3 then
print(`Mamar3(K): PaperZ2(L,K): where L is given by Hopefuls3(): Try:`):
print(`Mamar3(2000):`):


elif nargs=1 and args[1]=Mamar4 then
print(`Mamar4(K): PaperZ2(L,K): where L is given by Hopefuls4(): Try:`):
print(`Mamar4(2000):`):



elif nargs=1 and args[1]=MyIDs then
print(`MyIDs(C,F,F0,N): Given a constant C in decimals and another constant`):
print(`let's call it F, whose floating-point if F0`):
print(` and a positive integer N`):
print(`tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ`):
print(`MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);`):

elif nargs=1 and args[1]=NormalizePair then
print(`NoralizePair(P,K): Given a pair of sequenes of rational numbers P[1],P[2], outputs the`):
print(`sequence K^i*P[1][i], K^i*P[2][i]. Try:`):
print(`gu:=AnBn(1/2,0,0,1/2,50): NormalizePair([gu[1],gu[2]],8);`):
print(`gu:=AnBn(2/3,1/3,1/6,1/3,50): NormalizePair([gu[1],gu[2]],9/4);`):


elif nargs=1 and args[1]=Nu1 then
print(`Nu1(a,b,K,r): if you have a sequence of diophantine approximations A(n)+B(n)*c such that`):
print(`(i)|A(n)+B(n)*c|<=Cb^n, (ii) max(|A(n)|,|B(n)|)=OMEGA(a^n) (iii) K^n*dn(r*n)*A(n) and K^n*dn(r*n)*B(n) are integers`):
print(`outputs the implied rigorous delta (assuming that the divisibility lemmas are proved). Try:`):
print(`mu:=5*sqrt(5)/2-11/2; `):
print(`Nu1(1/mu,mu,16,4);`):
print(`Nu1(1/mu,mu,1,2);`):

elif nargs=1 and args[1]=OPEZ2 then
print(`OPEZ2(a1,a2,b1,b2,n,N): The second-order linear recurrence equation annihilated by the Generalized`):
print(`Beukers integral `):
print(`Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):`):
print(`Here N is the shift operator in n, Nf(n):=f(n+1).`):
print(`Try:`):
print(`For the original Apery Recurrence for Zeta(2) type: OPEZ2(0,0,0,0,n,N);`):
print(`For the Zudilin (2003) recurrence for Catalan's constant type: OPEZ2(1/2,0,0,1/2,n,N);`):
print(`For  a promising recurrence type `):
print(`OPEZ2(2/3,1/3,1/6,1/3,n,N);`):

elif nargs=1 and args[1]=OPEct then
print(`OPEct(c1,c2,n,N): The second-order linear recurrence equation with polynomial coefficients annihilating`):
print(`the contour integral Int(Int(((1+z)*(1+w)*(1+z+w))^n/(z^c1*w^c2),|z|=1,|w|=1). The case c1=1,c2=1 is Apery's`):
print(`Zeta(2) recurrence. Try:`):
print(`OPEct(1,1,n,N);`):
print(`OPEct(1/2,1/2,n,N);`):


elif nargs=1 and args[1]=OPEZ2g then
print(`OPEZ2g(a1,a2,b1,b2,c,n,N): the second-order linear recurrence operator annihilating the generalized Beukers sequence`):
print(`Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-a1)*(1-y)^(-a1)/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1);. Try:`):
print(`OPEZ2g(0,0,0,0,0,n,N);`):


elif nargs=1 and args[1]=PaperZ2 then
print(`PaperZ2(L,K): Given a list L of quadruples that are believed to be provably irrational and a large positive ineger K (around 2000 is OK)`):
print(` outputs a paper with sketches`):
print(`of proof. Try: `):
print(`PaperZ2([[0,0,0,0],[1/2,0,0,0]],2000):`):


elif nargs=1 and args[1]=PaperZ2g then
print(`PaperZ2g(L,K): Given a list L of pentuplues that are believed to be provably irrational and a large positive ineger K (around 2000 is OK)`):
print(` outputs a paper with sketches`):
print(`of proof. Try: `):
print(`PaperZ2g([[0,0,0,0,0],[0,0,0,0,1/2]],2000):`):

elif nargs=1 and args[1]=PPold then
print(`PPold(N): Inputs a positive integer N, outputs a list whose entries are`):
print(` (i): The largest prime that shows up, let's call it P. `):
print(`(ii) the list of lists whose j-th entry is the list primes between sqrt(p) and p with exponent is j.`):
print(`Try:`):
print(`PPold(LC(100));`):


elif nargs=1 and args[1]=PpG then
print(`PpG(a,b,n,m1,m2,C,M): Given rational numbers a and b between 0 and 1 and positive integers m1,m2 and n`):
print(`and a subset C in {1,...M-1} and an integer M`):
print(`it is the product of p between m1 and m2  such that frac{n/p} is between a and b`):
print(`and p mod M is in C. Try:`):
print(` PpG(0,1/2,1000,1000,2000,{0},1);`):

elif nargs=1 and args[1]=PpGlimit then
print(`The limit of log PpG(a,b,r1*n,r2*n,C,M)/n as n goes to infinity. Try:`):
print(`PpGlimit(1/3,2/3,3/2,3,{2},3);`):

elif nargs=1 and args[1]=PrimesF then
print(`PrimesF(N,n,k): inputs a HUGE integer N that is a product of small primes, and a much smaller integer n`):
print(`that generated N (via some process), outputs the list of length k, whose i-th entry is the list`):
print(`of primes>=sqrt(n) that show up with exponent i, followed by the list of primes that`):
print(`did not show up at all. Try:`):
print(`lu:=op(Hopefuls2()[3][1]): N:=CnDn(op(lu),2000)[1][2][-1],2000): PrimesF(N,2000,6);`):

elif nargs=1 and args[1]=PrintPpG then
print(`PrintPpG(P,n,L1,L2): P is the  same input as the P EvalPpG(P,n1) (q.v.) but n,L1 and L2 are symbols`):
print(`where n stands for n,  L1(n) stands for lcm(1...n) and L2(alpha,beta,a,b,C,M,n) stands for the`):
print(`product of all primes between a*n and b*n such that the fractional part of n/p is between alpha and beta`):
print(`and p mod M belongs to the set C where C is a subset of the set of nonneg. integers less than M relatively prime to it.`):
print(`If no modularity condition is present than M=1 and C={0}. Try:`):
print(``):
print(`For GBC(0,0,1/2,0) try: `):
print(`PrintPpG([ [[2,2],[1,0],[]],[[1, 1], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ]],n,LCM,PP);`):
print(``):
print(`For GBC(0,0,1/3,-2/3) try: `):
print(`PrintPpG([[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {2}, 3]]],[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$2, [0, 1/3, 0, 3/2, {0}, 1], [2/3, 1, 0, 3/2, {0}, 1]]]],n,LCM,PP);`);


elif nargs=1 and args[1]=rf then
print(`rf(a,n): The raising factorial a(a+1)...*(a+n-1). Try:`):
print(`rf(1/3,10);`):


elif nargs=1 and args[1]=Search then
print(`Search(N,K): All the 4-tuples [a,b,c,d]=[a1/N,a2/N,a3/N,a4/N] with the ai's beteen -(N-1) and N-1 such that`):
print(`the deltas for AnBn(a,b,c,d,e) seems to be positive. Try`):
print(`Search(2,100);`):

elif nargs=1 and args[1]=SeqFromRec then
print(`SeqFromRec(ope,n,N,ini,L): Given an (ordinary) recurrence operator`):
print(`ope(n,N) in the variable n and the shift operator N (Nf(n):=f(n+1))`):
print(`and given initial values [ini0,ini1,..,ini_{ORD-1}],computes`):
print(`the first L+1 terms of the sequence a(n) satisfying`):
print(`ope(n,N)a(n)=0 and a[i]=ini[i] for i=0,...,ORD-1`):
print(`For example, try: SeqFromRec(N^2-N-1,n,N,[0,1],10);`):


elif nargs=1 and args[1]=SortC then
print(`SortC(L,N,r): Given a list of 4-tuples, L, and a positive integer N, and a symbol r`):
print(`divides them into classes such that GBC(op(L[i])) are related to`):
print(`each other (conjecturally) by a fractional-linear transfomation with integers  less than N in absolute value`);
print(`The output is a list of lists such that the first entry is the main pentuple, followed by the symbol r[i],`):
print(`its decimal approximation, followed by the tuples, and the expression in terms of r[i]`):
print(`Try:`):
print(`SortC(Hopefuls5()[2],10000,r);`):

elif nargs=1 and args[1]=WadimHW then
print(`WadimHW(): A data-base of conjectured 3F2 identities (many of them probably equivalent via some transformation`):
print(`Homework for Wadim Zudilin to prove them`):

elif nargs=1 and args[1]=WadimHWv then
print(`WadimHWv(): A verbose version of WadimHW(): lots of conjectured evaluations. Each bunch are probably equivalent to each other`):
print(`via some transformation that Wadim Zudilin can figure out. Print:`):
print(`WadimHWv():`):


elif nargs=1 and args[1]=WadimHWvLP then
print(`WadimHWvLP(): Like WadimHWv() (q.v.), but with lprint insterad.`):
print(`WadimHWvLP():`):

elif nargs=1 and args[1]=TheoremZ2 then
print(`TheoremZ2(a1,a2,b1,b2,K,P): Inputs rational numbers a1,a2,b1,b2, and a large positive integer K and P either 0 `):
print(`if not known, or a proposed INTEGERrating factor, P`):
print(`Outputs a theorem regarding the constant`):
print(`IntGB(a1,a2,b1,b2,0) (q.v.)`):
print(`Either a suggested proof of irrationality or a way to compute it exponentially  fast.`):
print(`For the original Zeta(2), Try:`):
print(`TheoremZ2(0,0,0,0,2000, [[ [1,0],[1,0],[] ],[ [1,0],[1,2],[] ]]):`):
print(``):

print(`-----------------------`):
print(`The following examples give positive delta, i.e. yielding irrationality proofs`):
print(``):
print(`For GBC(0,0,1/2,0)  (alias 2*log(2)) try: `):
print(`TheoremZ2(0,0,1/2,0,2000,[ [[2,2],[1,0],[]],[[1, 1], [1, 0], [ [0,1/2,0,1,{0},1],  [1/2,1,0,2,{0},1]$2 ] ]]);`):
print(``):
print(`For GBC(0,0,1/3,-2/3)  (conjecturally -6+4*Pi/sqrt(3)), try: `):
print(`TheoremZ2(0,0,1/3,-2/3,2000,[[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {2}, 3]]],[[1, 0], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$2, [0, 1/3, 0, 3/2, {0}, 1], [2/3, 1, 0, 3/2, {0}, 1]]]]);`);

print(`For the equivalent GBC(0,-2/3,2/3,-2/3), conjecturally ( 45/8-5*sqrt(3)*Pi/6) but much simpler, try:`):
print(`TheoremZ2(0,-2/3,2/3,-2/3,2000,[[[1, 0], [1, 0], []],[[3, 3/2], [1, 0], [[0, 1/3, 0, 1, {0}, 1]]]]);`);


print(``):
print(`For GBC(-3/4,-3/4,-1/4,-1/4), conjecturally 92-64*sqrt(2),  try: `):
print(` TheoremZ2(-3/4,-3/4,-1/4,-1/4, 2000,[[[1, 0], [1, 0], []], [[2, 2], [1, 0], [[1/2, 3/4, 0, 2, {0}, 1]] ]],2000); `):



print(``):
print(`For GBC( -4/5, -4/5, -2/5, 2/5), conjecturally, -65/4+65/8*5^(1/2),  try: `):
print(` TheoremZ2( -4/5, -4/5, -2/5, 2/5, 2000,0); `):




print(``):
print(`For GBC(0, -1/2, 1/6, -1/2), (not yet identified, possibly not yet proved to be irrational) , also no INTEGERating factor yet, try: `):
print(` TheoremZ2(0,-1/2,1/6,-1/2,2000,0):`):

print(`----------------------------------`):


print(``):
print(`For GBC(  -5/6, -5/6, -1/2, -1/2), conjecturally, -1344/5+16384/105*3^(1/2) ,  try: `):
print(` TheoremZ2(   -5/6, -5/6, -1/2, -1/2, 2000,0); `):


print(``):
print(`For GBC(  -5/6, -5/6, -1/3, -2/3), conjecturally, 972/5*2^(2/3)-1536/5 ,  try: `):
print(` TheoremZ2( -5/6, -5/6, -1/3, -2/3, 2000,0); `):



print(``):
print(`For GBC( -2/3, -1/2, 1/2, -1/2 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(`Comment:The delta is very small, so this case is questionable`):
print(` TheoremZ2(-2/3, -1/2, 1/2, -1/2 , 3000,0); `):


print(``):
print(`For GBC(  -6/7, -6/7, -4/7, -5/7 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2(  -6/7, -6/7, -4/7, -5/7, 2000,0); `):



print(``):
print(`For GBC( -6/7, -6/7, -4/7, 4/7), conjecturally`):
print(`the root of -110592*x^3+1225728*x^2+45000816*x-101163391=0 `):
print(`or, using Cardano`):
print(`-665/576*(28+84*I*3^(1/2))^(1/3)-4655/144/(28+84*I*3^(1/2))^(1/3)+133/36-133/96*I*3^(1/2)*(5/6*(28+84*I*3^(1/2))^(1/3)-70/3/(28+84*I*3^(1/2))^(1/3))`):
print(`type: `):
print(` TheoremZ2(  -6/7, -6/7, -4/7, 4/7, 2000,0); `):


print(``):
print(`For GBC( -6/7, -5/7, -3/7, -5/7 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2( -6/7, -5/7, -3/7, -5/7, 2000,0); `):


print(``):
print(`For GBC( -6/7, -5/7, -2/7, -1/7 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2( -6/7, -5/7, -2/7, -1/7, 2000,0); `):


print(``):
print(`For GBC( -6/7, -4/7, -1/7, -1/7  ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2( -6/7, -4/7, -1/7, -1/7, 2000,0); `):


print(``):
print(`For GBC((-6/7, -3/7, -5/7, -3/7 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2(-6/7, -3/7, -5/7, -3/7, 2000,0); `):


print(``):
print(`For GBC(  -6/7, -1/7, 4/7, 2/7), conjecturally`):
print(`the root of  2299968*x^3+7074144*x^2-11234916*x-12663217=0 `):
print(`or, using Cardano`):
print(`245/792*(-28+84*I*3^(1/2))^(1/3)+1715/198/(-28+84*I*3^(1/2))^(1/3)-203/198`):
print(`type: `):
print(` TheoremZ2( -6/7, -1/7, 4/7, 2/7, 2000,0); `):


print(``):
print(`For GBC( -5/7, -3/7, -4/7, -2/7 ), not yet identified, also no INTEGERating factor yet ,  try: `):
print(` TheoremZ2( -5/7, -3/7, -4/7, -2/7, 2000,0); `):
print(``):
print(`----------------------------------------------------`):
print(`The following examples are for constants that yield negative deltas, but it is still interesting`):
print(``):
print(`For GBC(1/2,-1/2,-1/2,1/2), (conjecturally 4-8/Pi) , type:`):
print(`TheoremZ2(1/2,-1/2,-1/2,1/2,2000,[[[1, 0], [1, 0], []],[[2, 4], [1, 0], [[0, 1/2, 0, 1, {0}, 1]]]]);`);


print(`For GBC(1/2,0,0,1/2), twice the Catalan constant , type:`):
print(`TheoremZ2(1/2,0,0,1/2,2000,[[[1, 0], [1, 0], []],[[2, 4], [2, 2], [ ]]]);`);


print(`For GBC(1/3,0,0,1/3), whatever it is , type:`):
print(`TheoremZ2(1/3,0,0,1/3,2000,[[[1, 0], [1, 0], []],[[3,3], [3/2, 2], [  [1/3,2/3,3/2,3,{2},3]$2     ]    ]]);`);

elif nargs=1 and args[1]=TheoremZ2g then
print(`TheoremZ2g(a1,a2,b1,b2,e,K): Inputs rational numbers a1,a2,b1,b2, and e, and a large positive integer K and outputs`):
print(`a theorem regarding the constant`):
print(`IntGBg(a1,a2,b1,b2,e,0) (q.v.)`):
print(`Either a suggested proof of irrationality or a way to compute it exponentially  fast.`):
print(`Try:`):
print(`TheoremZ2g(0,0,0,0,0,2000):`):


elif nargs=1 and args[1]=TheoremZ2gnum then
print(`TheoremZ2gnum(a1,a2,b1,b2,e,K,num): Like  TheoremZ2g(a1,a2,b1,b2,e,K) (q.v.) but as a numbered theorem. Try:`):
print(`TheoremZ2gnum(0,0,0,0,0,2000,3):`):


elif nargs=1 and args[1]=TheoremZ2num then
print(`TheoremZ2num(a1,a2,b1,b2,K,num): Like TheoremZ2num(a1,a2,b1,b2,K) but a numbered theorem `):
print(`Try:`):
print(`TheoremZ2num(0,0,0,0,2000,1):`):

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

fi:


end:


Digits:=100:


#start pre-computed


#Hopefuls3(): A lis of 35 tuples [a1,a2,b1,b2] for which there is a potenial irrationality proof AND
#for GBC(a1,a2,b1,b2) (q.v.) and Maple can't identitify in terms of algebraic and/or combinations of log
#hence form candiates for a NEW irrationality theorem for a natural  constant.Try:
#Hopefuls3();
Hopefuls3:=proc():
[[5/3, 5/3, 0, 0], [2/3, 5/3, 0, 0], [4/3, 4/3, 0, 0], [2/3, 2/3, 0, 0], [4/3, 1/3, 0, 0], [1/3, 2/3, 1/2, 0], [2/3, 1/3, 1/2, 0], [1/3, 2/3, 3/2, 0], [2/3, 1/3, 3/
2, 0], [1/3, 2/3, 5/2, 0], [1/3, 0, 5/3, 0], [1/2, 1/2, 5/3, 0], [2/3, 0, 4/3, 0], [1/2, 1/2, 4/3, 0], [1/2, 1/2, 2/3, 0], [1/2, 1/2, 1/3, 0], [1/3, 2/3, 1/2, 1/2],
[2/3, 1/3, 1/2, 1/2], [1/3, 1/6, 1/2, 1/2], [1/2, 1/6, 2/3, 1/3], [1/7, 3/7, 5/7, 5/7], [3/7, 3/7, 4/7, 5/7], [2/7, 4/7, 3/7, 5/7], [4/7, 1/7, 3/7, 5/7], [2/7, 3/7,
1/7, 5/7], [2/7, 4/7, 1/7, 4/7], [3/7, 2/7, 4/7, 4/7], [5/7, 1/7, 4/7, 4/7], [3/7, 1/7, 5/7, 4/7], [4/7, 2/7, 5/7, 3/7], [2/7, 1/7, 5/7, 3/7], [4/7, 2/7, 1/7, 2/7],
[5/7, 1/7, 2/7, 2/7], [1/7, 1/7, 3/7, 2/7], [2/7, 1/7, 4/7, 1/7]]:
end:

#Hopefuls4(): A lis of 88 tuples [a1,a2,b1,b2] for which there is a potenial irrationality proof AND
#for GBC(a1,a2,b1,b2) (q.v.) and Maple CAN identitify in terms of algebraic and/or combinations of log
#hence there already exists a proof of irrationality but there is a potential for improved irrationailty measure
Hopefuls4:=proc():
[[0, 0, 0, 0], [1/2, 0, 0, 0], [3/2, 0, 0, 0], [5/2, 0, 0, 0], [1/3, 1/3, 0, 0], [1/3, 0, 2/3, 0], [2/3, 1/3, 5/6, 1/2], [5/6, 1/6, 2/3, 1/2], [1/6, 2/3, 1/2, 1/2],
[5/6, 1/3, 1/2, 1/2], [1/6, 1/6, 1/2, 1/2], [5/6, 5/6, 1/2, 1/2], [1/6, 5/6, 1/3, 1/2], [1/3, 2/3, 1/6, 1/2], [1/4, 3/4, 5/4, 1/2], [3/4, 1/4, 3/4, 1/2], [1/4, 3/4,
1/4, 1/2], [1/6, 5/6, 4/3, 1/2], [5/6, 1/6, 5/3, 1/2], [2/3, 1/3, 5/6, 3/2], [5/6, 1/6, 2/3, 3/2], [1/6, 5/6, 1/3, 3/2], [1/3, 2/3, 1/6, 3/2], [1/4, 3/4, 5/4, 3/2],
[3/4, 1/4, 3/4, 3/2], [1/4, 3/4, 1/4, 3/2], [5/6, 1/6, 5/3, 5/3], [1/6, 5/6, 2/3, 5/3], [5/6, 1/6, 2/3, 5/3], [1/3, 2/3, 1/6, 5/3], [1/3, 2/3, 5/6, 5/3], [2/3, 1/3,
5/6, 4/3], [1/6, 5/6, 1/3, 4/3], [5/6, 1/6, 1/3, 4/3], [2/3, 1/3, 1/6, 4/3], [5/6, 1/6, 4/3, 4/3], [1/6, 5/6, 4/3, 4/3], [1/6, 5/6, 3/2, 4/3], [1/3, 2/3, 5/6, 2/3],
[1/6, 5/6, 2/3, 2/3], [5/6, 1/6, 2/3, 2/3], [5/6, 1/6, 1/2, 2/3], [5/6, 5/6, 1/3, 2/3], [1/6, 2/3, 1/3, 2/3], [1/6, 5/6, 5/3, 2/3], [5/6, 1/6, 5/3, 2/3], [1/6, 5/6,
1/2, 1/3], [1/6, 5/6, 3/2, 1/3], [1/6, 5/6, 5/2, 1/3], [1/6, 5/6, 4/3, 1/3], [5/6, 1/6, 4/3, 1/3], [5/6, 1/3, 2/3, 1/3], [1/6, 1/3, 2/3, 1/3], [1/6, 1/6, 2/3, 1/3],
[5/6, 1/6, 1/3, 1/3], [1/6, 5/6, 1/3, 1/3], [3/4, 1/4, 5/4, 1/4], [1/4, 1/4, 3/4, 1/4], [5/4, 5/4, 3/4, 1/4], [1/4, 5/4, 3/4, 1/4], [1/4, 3/4, 3/4, 3/4], [2/5, 2/5,
1/5, 4/5], [3/5, 4/5, 1/5, 4/5], [2/5, 3/5, 4/5, 4/5], [4/5, 1/5, 3/5, 3/5], [1/5, 3/5, 2/5, 3/5], [1/5, 1/5, 3/5, 2/5], [4/5, 2/5, 3/5, 2/5], [2/5, 1/5, 4/5, 1/5],
[1/2, 1/6, 5/6, 1/6], [1/3, 1/6, 5/6, 1/6], [4/3, 1/6, 5/6, 1/6], [5/2, 1/6, 5/6, 1/6], [3/2, 1/6, 5/6, 1/6], [1/6, 5/6, 2/3, 5/6], [1/6, 5/6, 1/2, 5/6], [5/3, 5/6,
1/6, 5/6], [3/2, 5/6, 1/6, 5/6], [3/7, 3/7, 2/7, 5/7], [1/7, 4/7, 2/7, 5/7], [4/7, 3/7, 1/7, 5/7], [3/7, 4/7, 2/7, 4/7], [1/7, 1/7, 3/7, 4/7], [5/7, 2/7, 4/7, 4/7],
[4/7, 3/7, 5/7, 3/7], [2/7, 1/7, 4/7, 3/7], [5/7, 2/7, 1/7, 2/7], [3/7, 1/7, 5/7, 2/7]]:
end:



Hopefuls5:=proc():
[
[[-1/2, -1/2, 0, -1/2], [-1/2, -1/2, 0, 0], [-1/2, 0, 0, 0], [-1/2, 1/2, 0, -1/
2], [-1/2, 1/2, 0, 0], [0, -1/2, 1/2, -1/2], [0, 0, 1/2, -1/2], [0, 0, 1/2, 0],
[0, 1/2, 1/2, -1/2]],
[[-2/3, -2/3, -1/3, -1/3], [-2/3, -2/3, 0, 0], [-2/3, -1/3, 0, -1/3], [-2/3, -1
/3, 0, 2/3], [-2/3, 0, -1/3, 0], [-2/3, 0, 2/3, 0], [-2/3, 1/3, -1/3, -1/3], [-\
2/3, 1/3, 0, 0], [-2/3, 2/3, 0, -1/3], [-2/3, 2/3, 0, 2/3], [-1/3, -2/3, 0, -2/
3], [-1/3, -2/3, 0, 1/3], [-1/3, -1/3, 0, 0], [-1/3, -1/3, 1/3, -2/3], [-1/3, 0
, 1/3, 0], [-1/3, 1/3, 0, -2/3], [-1/3, 1/3, 0, 1/3], [-1/3, 2/3, 0, 0], [-1/3,
2/3, 1/3, -2/3], [0, -2/3, 2/3, -2/3], [0, -1/3, 1/3, -1/3], [0, 0, 1/3, -2/3],
[0, 0, 1/3, 1/3], [0, 0, 2/3, -1/3], [0, 0, 2/3, 2/3], [0, 1/3, 2/3, -2/3], [0,
2/3, 1/3, -1/3], [1/3, -2/3, 2/3, -1/3], [1/3, 0, 2/3, 0], [1/3, 1/3, 2/3, -1/3
]],
[[-3/4, -3/4, -1/4, -3/4], [-3/4, -3/4, -1/4, -1/4], [-3/4, -3/4, -1/4, 1/4], [
-3/4, -3/4, -1/4, 3/4], [-3/4, -3/4, 3/4, -3/4], [-3/4, -3/4, 3/4, -1/4], [-3/4
, -3/4, 3/4, 1/4], [-3/4, -3/4, 3/4, 3/4], [-3/4, -1/2, -3/4, -1/4], [-3/4, -1/
2, -3/4, 3/4], [-3/4, -1/2, 1/4, -1/4], [-3/4, -1/2, 1/4, 3/4], [-3/4, -1/4, -3
/4, 1/2], [-3/4, -1/4, -1/4, -1/4], [-3/4, -1/4, -1/4, 3/4], [-3/4, -1/4, 1/4,
-1/2], [-3/4, -1/4, 1/4, 1/2], [-3/4, -1/4, 3/4, -1/4], [-3/4, -1/4, 3/4, 3/4],
[-3/4, 1/4, -1/4, -3/4], [-3/4, 1/4, -1/4, -1/4], [-3/4, 1/4, -1/4, 1/4], [-3/4
, 1/4, 3/4, -3/4], [-3/4, 1/4, 3/4, -1/4], [-3/4, 1/4, 3/4, 1/4], [-3/4, 1/2, -\
3/4, 3/4], [-3/4, 1/2, 1/4, -1/4], [-3/4, 1/2, 1/4, 3/4], [-3/4, 3/4, -1/4, -1/
4], [-3/4, 3/4, -1/4, 3/4], [-3/4, 3/4, 1/4, -1/2], [-3/4, 3/4, 1/4, 1/2], [-3/
4, 3/4, 3/4, -1/4], [-3/4, 3/4, 3/4, 3/4], [-1/2, -3/4, -1/2, -1/4], [-1/2, -3/
4, -1/2, 3/4], [-1/2, -3/4, 1/2, -1/4], [-1/2, -3/4, 1/2, 3/4], [-1/2, -1/4, -1
/2, 1/4], [-1/2, -1/4, 1/2, -3/4], [-1/2, -1/4, 1/2, 1/4], [-1/2, 1/4, 1/2, -1/
4], [-1/2, 3/4, 1/2, -3/4], [-1/4, -3/4, -1/4, -1/2], [-1/4, -3/4, -1/4, 1/2],
[-1/4, -3/4, 1/4, -3/4], [-1/4, -3/4, 1/4, 1/4], [-1/4, -3/4, 3/4, -1/2], [-1/4
, -3/4, 3/4, 1/2], [-1/4, -1/2, -1/4, 1/4], [-1/4, -1/2, 3/4, -3/4], [-1/4, -1/
2, 3/4, 1/4], [-1/4, -1/4, 1/4, -3/4], [-1/4, -1/4, 1/4, -1/4], [-1/4, -1/4, 1/
4, 1/4], [-1/4, -1/4, 1/4, 3/4], [-1/4, 1/4, -1/4, 1/2], [-1/4, 1/4, 1/4, -3/4]
, [-1/4, 1/4, 1/4, 1/4], [-1/4, 1/4, 3/4, -1/2], [-1/4, 1/4, 3/4, 1/2], [-1/4,
1/2, 3/4, -3/4], [-1/4, 1/2, 3/4, 1/4], [-1/4, 3/4, 1/4, -3/4], [-1/4, 3/4, 1/4
, -1/4], [-1/4, 3/4, 1/4, 3/4], [1/4, -3/4, 3/4, -3/4], [1/4, -3/4, 3/4, -1/4],
[1/4, -3/4, 3/4, 1/4], [1/4, -3/4, 3/4, 3/4], [1/4, -1/2, 1/4, -1/4], [1/4, -1/
2, 1/4, 3/4], [1/4, -1/4, 1/4, 1/2], [1/4, -1/4, 3/4, -1/4], [1/4, -1/4, 3/4, 3
/4], [1/4, 1/4, 3/4, -3/4], [1/4, 1/4, 3/4, -1/4], [1/4, 1/4, 3/4, 1/4], [1/4,
1/2, 1/4, 3/4], [1/4, 3/4, 3/4, -1/4], [1/4, 3/4, 3/4, 3/4], [1/2, -3/4, 1/2, -\
1/4], [1/2, -3/4, 1/2, 3/4], [1/2, -1/4, 1/2, 1/4], [3/4, -3/4, 3/4, -1/2], [3/
4, -3/4, 3/4, 1/2], [3/4, -1/2, 3/4, 1/4], [3/4, 1/4, 3/4, 1/2]],
[[-4/5, -4/5, -2/5, -3/5], [-4/5, -4/5, -2/5, 2/5], [-4/5, -4/5, -1/5, -1/5], [
-4/5, -4/5, -1/5, 4/5], [-4/5, -4/5, 3/5, -3/5], [-4/5, -4/5, 3/5, 2/5], [-4/5,
-4/5, 4/5, -1/5], [-4/5, -4/5, 4/5, 4/5], [-4/5, -3/5, -1/5, -2/5], [-4/5, -3/5
, -1/5, 3/5], [-4/5, -3/5, 4/5, -2/5], [-4/5, -3/5, 4/5, 3/5], [-4/5, -2/5, -3/
5, -2/5], [-4/5, -2/5, -3/5, 3/5], [-4/5, -2/5, 2/5, -2/5], [-4/5, -2/5, 2/5, 3
/5], [-4/5, -1/5, -3/5, -3/5], [-4/5, -1/5, -3/5, 2/5], [-4/5, -1/5, -2/5, -1/5
], [-4/5, -1/5, -2/5, 4/5], [-4/5, -1/5, 2/5, -3/5], [-4/5, -1/5, 2/5, 2/5], [-\
4/5, -1/5, 3/5, -1/5], [-4/5, -1/5, 3/5, 4/5], [-4/5, 1/5, -2/5, -3/5], [-4/5,
1/5, -2/5, 2/5], [-4/5, 1/5, -1/5, -1/5], [-4/5, 1/5, 3/5, -3/5], [-4/5, 1/5, 3
/5, 2/5], [-4/5, 1/5, 4/5, -1/5], [-4/5, 2/5, -1/5, -2/5], [-4/5, 2/5, 4/5, -2/
5], [-4/5, 3/5, -3/5, -2/5], [-4/5, 3/5, -3/5, 3/5], [-4/5, 3/5, 2/5, -2/5], [-\
4/5, 3/5, 2/5, 3/5], [-4/5, 4/5, -3/5, -3/5], [-4/5, 4/5, -3/5, 2/5], [-4/5, 4/
5, -2/5, -1/5], [-4/5, 4/5, -2/5, 4/5], [-4/5, 4/5, 2/5, -3/5], [-4/5, 4/5, 2/5
, 2/5], [-4/5, 4/5, 3/5, -1/5], [-3/5, -4/5, -1/5, -4/5], [-3/5, -4/5, -1/5, 1/
5], [-3/5, -4/5, 4/5, -4/5], [-3/5, -4/5, 4/5, 1/5], [-3/5, -3/5, -2/5, -2/5],
[-3/5, -3/5, -2/5, 3/5], [-3/5, -3/5, 1/5, -1/5], [-3/5, -3/5, 1/5, 4/5], [-3/5
, -3/5, 3/5, -2/5], [-3/5, -3/5, 3/5, 3/5], [-3/5, -2/5, -1/5, -1/5], [-3/5, -2
/5, -1/5, 4/5], [-3/5, -2/5, 1/5, -2/5], [-3/5, -2/5, 1/5, 3/5], [-3/5, -2/5, 4
/5, -1/5], [-3/5, -2/5, 4/5, 4/5], [-3/5, -1/5, -2/5, -4/5], [-3/5, -1/5, -2/5,
1/5], [-3/5, -1/5, 3/5, -4/5], [-3/5, -1/5, 3/5, 1/5], [-3/5, 1/5, -1/5, -4/5],
[-3/5, 1/5, -1/5, 1/5], [-3/5, 1/5, 4/5, -4/5], [-3/5, 1/5, 4/5, 1/5], [-3/5, 2
/5, -2/5, -2/5], [-3/5, 2/5, 1/5, -1/5], [-3/5, 2/5, 1/5, 4/5], [-3/5, 2/5, 3/5
, -2/5], [-3/5, 3/5, -1/5, -1/5], [-3/5, 3/5, -1/5, 4/5], [-3/5, 3/5, 1/5, -2/5
], [-3/5, 3/5, 1/5, 3/5], [-3/5, 3/5, 4/5, -1/5], [-3/5, 4/5, -2/5, -4/5], [-3/
5, 4/5, 3/5, -4/5], [-2/5, -4/5, 2/5, -1/5], [-2/5, -4/5, 2/5, 4/5], [-2/5, -3/
5, -1/5, -3/5], [-2/5, -3/5, -1/5, 2/5], [-2/5, -3/5, 1/5, -4/5], [-2/5, -3/5,
1/5, 1/5], [-2/5, -3/5, 4/5, -3/5], [-2/5, -3/5, 4/5, 2/5], [-2/5, -2/5, -1/5,
-4/5], [-2/5, -2/5, -1/5, 1/5], [-2/5, -2/5, 2/5, -3/5], [-2/5, -2/5, 2/5, 2/5]
, [-2/5, -2/5, 4/5, -4/5], [-2/5, -2/5, 4/5, 1/5], [-2/5, -1/5, 1/5, -1/5], [-2
/5, -1/5, 1/5, 4/5], [-2/5, 1/5, 2/5, -1/5], [-2/5, 2/5, -1/5, -3/5], [-2/5, 2/
5, -1/5, 2/5], [-2/5, 2/5, 1/5, -4/5], [-2/5, 2/5, 1/5, 1/5], [-2/5, 2/5, 4/5,
-3/5], [-2/5, 2/5, 4/5, 2/5], [-2/5, 3/5, -1/5, -4/5], [-2/5, 3/5, -1/5, 1/5],
[-2/5, 3/5, 2/5, -3/5], [-2/5, 3/5, 4/5, -4/5], [-2/5, 3/5, 4/5, 1/5], [-2/5, 4
/5, 1/5, -1/5], [-2/5, 4/5, 1/5, 4/5], [-1/5, -4/5, 2/5, -4/5], [-1/5, -4/5, 2/
5, 1/5], [-1/5, -4/5, 3/5, -2/5], [-1/5, -4/5, 3/5, 3/5], [-1/5, -3/5, 3/5, -3/
5], [-1/5, -3/5, 3/5, 2/5], [-1/5, -2/5, 1/5, -3/5], [-1/5, -2/5, 1/5, 2/5], [-\
1/5, -1/5, 1/5, -4/5], [-1/5, -1/5, 1/5, 1/5], [-1/5, -1/5, 2/5, -2/5], [-1/5,
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]:
end:


AlmostHopefuls1G:=proc():[[0, 0, 1/3, 0, 1/2], [1/3, 0, 0, 0, 1/2],  [2/3, 2/3, 0, 1/2, 1/2], [1/3, 1/3, 0, 1/2, 1/2]]:end:

Hopefuls1G:=proc():
[
[2/3, 0, 0, 0, 1/2], [0, 0, 0, 0, 1/2]
#,[1/2, 0, 1/2, 0, 1/2]
]:
end:

Hopefuls1:=proc():

[[[0, 0, 0, 0], .160376328282995365630650037915397621728928601629473517662941887093415487440248087647248677678121518, 1/6*Pi^2, 1/6*Pi^2], [[1/2, 0, 0, 0], .2154464\
06424598151262168999617658868512485219126761727588124913699012907249772226740948093718322976, arcsinh(15/8), 2*ln(2)], [[3/2, 0, 0, 0], .166836981627750005879354379\
653292869850893026629198799852660714262138415414482514946546100577966226, 2/3-2/3*ln(2), (-4/3+4/3*ln(2))*undefined], [[5/2, 0, 0, 0], .1231202465221477609631440042\
35422251425266242442746382143772502329776045148689719343663790627405749, 8/5-6/5*ln(2), (-16/15+4/5*ln(2))*undefined], [[5/3, 5/3, 0, 0], .1089758980846213061309503\
58824770117731793593303480987855061282057214950716954422285424878997274086, 6.28931910992790499083170956069901802519769274732983680579783890954337245357689543715777\
4983496884236], [[2/3, 5/3, 0, 0], .79427820438207856136462885505148001158517349587376950111071029851387999533856968645044703607043207e-1, 8.12759872846843570118815\
6515284311464568132496185481151139769870776246362252707767368249976424120338], [[4/3, 4/3, 0, 0], .13425814677987284161587612658217116568237660245803757559430401384\
1294598691826664978175406982941537, -3.392249205292772313242597822052694665355082810115925719462356169235153976407942354605156235265075211], [[2/3, 2/3, 0, 0], .\
77555472942058299294890245772734397686394140727445794410953968480254870078278865225780393768078929e-1, 2.40689968211710892529703912882107786614203312404637028778494\
2467694061590563176941842062494106030084], [[4/3, 1/3, 0, 0], .80292413538282695578209820858592042607139681906615304682354872731858098975614075589518997038285761e-1
, -.1568996821171089252970391288210778661420331240463702877849424676940615905631769418420624941060300844], [[1/3, 1/3, 0, 0], .1145078921418362632389713583308245678\
91283265654490297402507477702947791834722240260568005653457960, 1/3*Pi*3^(1/2)], [[2/3, 2/3, 1/2, 0], .\
8534813295373098656678565432808812260389557801171205987199801353840375831777527247459289326419961e-2, 1.778880661310756710839363682262016180191100844089420639623266\
647525728758775766255861041472749498289], [[1/3, 2/3, 1/2, 0], .62063859138631951918830664834530216548277442892012316711029498246954617281970843278608631097812307e-\
1, 2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177, 1/6*Pi^(3/2)*(I*3^(1/2)-1)/GAMMA(2/3)/GAMMA(5/6)*3^(1/2)],
[[2/3, 1/3, 1/2, 0], .8512659681647715474312113899794985919242782480448555885110711696075396491135209415458676916925656e-2, 1.29355477961489526747675751256560581882\
8929257420200788177626770763033747278022225220322143942310013, -1/4/Pi^(1/2)*(3*I+3^(1/2))*GAMMA(2/3)*GAMMA(5/6)*3^(1/2)], [[1/3, 1/3, 1/2, 0], .\
48012102379455875774226363652309251068540896340614084544482158212964711829988469139773884354234550e-1, 1.47458599237119248035285239560458739592183828655305837044573\
7770225992212437276431256560550559864208], [[4/3, 1/3, 1/2, 0], .18680612933934640881521627991766476906237971272327613170999879798523448383836634147866754275116701e\
-1, .3050828015257615039294295208790825208156323426893883259108524459548015575125447137486878898880271584, 0], [[4/3, 4/3, 1/2, 0], .\
31336975241847688066933897563300305132550466500373672720124119657761889067955042565983042423133728e-1, -.96610509732991736812349833846160558857264340029357042965600\
82195790972743530467509397961926959524728, 0], [[1/3, 4/3, 1/2, 0], .\
35168026675768535702082519401691662975126384606441057926017505154838115268995029776632386043495126e-1, -2.2372929961855962401764261978022936979609191432765291852228\
68885112996106218638215628280275279932104, 0], [[2/3, 5/3, 1/2, 0], .\
6679648097534732947244127066457732958062515685180661814001618112624338974324666003029695655969177e-2, 3.697201653276891777098409205655040450477752110223551599058166\
618814321896939415639652603681873745722], [[5/3, 5/3, 1/2, 0], .8450134528211736716720029799132886149040764182330178136736577944220389780519724660353963406536421e-2
, 3.333515584717557824863035804443246068232519515175647684974273771925538633309540859727045750043657353], [[2/3, 2/3, 3/2, 0], .\
45990725461055595377622290656752554054821245125554968054987075854175574220814657266791051981828482e-1, -.22111933868924328916063631773798381980889915591057936037673\
33524742712412242337441389585272505017113, 0], [[1/3, 2/3, 3/2, 0], .\
18870440581602688763698192575873349249798362219652008617619083406016692194631613644919279551255572e-1, -.70109105266272713058750953952514706773151110271199304809099\
69935381423829909016480354261332425123924, 0], [[5/3, 2/3, 3/2, 0], .\
16540650478429310245595015565837977054287130735245043381363016449519786076949793719899994098430361e-1, -2.5577613226215134216787273645240323603822016881788412792465\
33295051457517551532511722082945498996577, 0], [[2/3, 1/3, 3/2, 0], .\
11523429780165650732046049820208170280048644329061030500497508618642172149798877315665719267672350e-1, .431184926538298422492252504188535272942976419140066929392542\
2569210112490926740750734407146474366711, 0], [[1/3, 1/3, 3/2, 0], .\
54733959194415774208096046415347947740989586187176376185306428594787540862459911769197136008739405e-1, .105082801525761503929429520879082520815632342689388325910852\
4459548015575125447137486878898880271584, 0], [[1/4, 3/4, 3/2, 0], .\
6174857123506901740155162188103229385598092139607246883164926147559532209261619649834527891352606e-2, -1.31102877714605990523241979494555970684137747571581158140841\
0851900395293535207125115147766480714547, 0], [[4/3, 4/3, 3/2, 0], .\
57080763330935117519183367972079316473647113109225680664109708869084027444133087847864104691324432e-1, 2.11864649809279812008821309890114684898045957163826459261143\
4442556498053109319107814140137639966052, 0], [[1/3, 4/3, 3/2, 0], .\
34275055268555036020870920918514230647635686698917094049688031452955719945886036230260969748758443e-1, 1.76270700381440375982357380219770630203908085672347081477713\
1114887003893781361784371719724720067896, 0], [[5/3, 5/3, 3/2, 0], .\
21759061056958017821726757242898101766801345014922488002063944522713901155567698900776251265988309e-1, -.84860082663844588854920460282752022523887605511177579952908\
33094071609484697078198263018409368728609], [[2/3, 2/3, 5/2, 0], .14050745296496346514632849177894226050585170283278969403347674682346832099385731844785224576001215\
e-1, -.7595414400483784767727921094385367567333553340940986293787143267305812311947865945834825583939322384, 0], [[1/3, 2/3, 5/2, 0], .\
12178173924175540179499094164125184835472752517283072929116754854283575503928909719683401277717602e-1, -1.6358791228796966380375222588920098247068592396613171122123\
26318255665560312103845415994310899195582, 0], [[5/3, 2/3, 5/2, 0], .\
56636174555062574158713747846282923813256178257386925996443707453446303764852838939142798841732059e-1, -.32671603213545973496381790642790291885339493546347616226040\
01148456274473454024648337511635030102676, 0], [[1/3, 1/3, 5/2, 0], .\
19707640313118494251578220072897380940583583804075211535369098479962831781699489252835565318632870e-1, .382430802912817416592547267132793903375298108770650440375263\
7604591666097966762717020405170589609388, 0], [[4/3, 1/3, 5/2, 0], .\
69946056723345908140947640928532606751140544878731880027597294663796961111599788701709552108372052e-1, .684751595422715488211711437362752437553102971931835022267442\
6621355953274623658587539363303359185248, 0], [[4/3, 4/3, 5/2, 0], .\
65367811334792290037305298408907280761189255679704928025861618518240588395834638782133374259129591e-1, 3.64406050572160563973536070329655945305862128508520622216569\
6672330505840672042676557579587080101844, 0], [[1/3, 4/3, 5/2, 0], .\
38204073319455351399453963321802274048608434070910337416235959140843481279430626723926100308021977e-1, 6.60336941602049579125900996923036646856413959823857742206395\
0682525416353881719494361222843824285163, 0], [[2/3, 5/3, 5/2, 0], .\
25867528860838575676510557352070258583606873440621761696823001876370090741634096234339620150033897e-1, 4.90839504016932466870477238303487864856674366932934520282550\
0143557034309181753081042188954378762835], [[1/3, 0, 5/3, 0], .64102958178967597786497191815391791080609952668713253490321907401019356059545234156874039416987197e-1
, -.1045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896, undefined], [[1/2, 1/2, 5/3, 0], .\
52122665489671418843991262359262154823048120605590975603982469306691909746901768423497191156444029e-1, -.70109105266272713058750953952514706773151110271199304809099\
69935381423829909016480354261332425123924, 0], [[2/3, 0, 4/3, 0], .\
77440410652469971117584683804280108413696392053447793078410426809700374570167977158676255031614567e-1, .802299894039036308432346376273692622047344374682123429261647\
4892313538635210589806140208313686766948, undefined], [[1/2, 1/2, 4/3, 0], .\
12856811824821736978725300168073003933869095642604059788924795180518596409583120933356246521283485e-1, .431184926538298422492252504188535272942976419140066929392542\
2569210112490926740750734407146474366711, 0], [[1/6, 1/2, 4/3, 0], .\
12984609452726757122041483674445629629511706008452306237888465232560736585804960363983501094504379e-1, .254235667938134586607857934065902100679693618907823604925710\
3716290012979271205947905732415733559653, 0], [[1/6, 5/2, 4/3, 0], .\
65043690264544996479894720895647503341740418256130637457473802943127353269191373420097568102244741e-1, .809791223493690142163413489621457656235249174463379160481265\
9271845568534826761503461287971289115209], [[1/6, 3/2, 4/3, 0], .35518757216316468304243496914113076327135737713086554390358414710217948261033626829080070764598143e\
-1, 1.175138002542935839882382534798470868026053904482313876518087409924669262520907856247813149813378597], [[1/3, 0, 2/3, 0], .114278625461300786643597068627770913\
945086731959330099720885746739255139009680041134955153344236286, 2/9*Pi*3^(1/2), 2/9*Pi*3^(1/2)], [[1/2, 1/2, 2/3, 0], .\
62019395540823614173051823407943690646092589431996281485773318842507493943248709915281940163956735e-1, 1.40218210532545426117501907905029413546302220542398609618199\
3987076284765981803296070852266485024785, 2/9*Pi^(3/2)*3^(1/2)/GAMMA(2/3)/GAMMA(5/6)], [[5/6, 1/2, 2/3, 0], .\
8131019265800217346393011915162586052748491018350714252085301640430742944386124181481740744537750e-2, 1.185920440873837807226242454841344120127400562726280426415511\
098350485839183844170574027648499665526], [[5/6, 5/2, 2/3, 0], .39922157926305418069715192242242940513709593592828940505256517963405749704301303108340999519486807e-\
1, 1.909245546629038962510050581547804744762945851260145510483222190320659042404054870879513565693608259], [[5/6, 3/2, 2/3, 0], .\
14705939933339902485200491031644567373103480636573951587543933589664191086721972610694895422988836e-1, 1.70517421508100894778581824301602157358813445878589418616435\
5530034305011701021674481388630332664385], [[1/2, 1/2, 1/3, 0], .41554337044461010679739525540818002650076063684313827581877601612321049046547493046060440602080968e\
-1, 1.724739706153193689969010016754141091771905676560267717570169027684044996370696300293762858589746685, 2*GAMMA(2/3)/Pi^(1/2)*GAMMA(5/6)], [[1/6, 1/2, 1/3, 0], .\
48026111905934226065219208445225199690060840594318543845575064155998637573396681555726998016093023e-1, 1.96611465649492330713713652747278319456245104873741116059431\
7026967989616583035241675414067413152277], [[1/6, 5/2, 1/3, 0], .83993645601174156781426604438713258754019661066659679426532950124870895615547740214884518990793780e\
-1, -7.043340424100957725385384166056677500236013208585933432299745165047090344277181594461939245937415666], [[1/6, 1/6, 0, 1/2], .\
48167739163119469339031621660296642063957659582516942459984651114970862753073093266953863081352579e-1, 2.94917198474238496070570479120917479184367657310611674089147\
5540451984424874552862513121101119728416], [[2/3, 1/3, 5/6, 1/2], .\
50650410384634165913813072620628502426328782155099083611022322798493659258463300786019388908401531e-1, 2^(2/3)], [[5/6, 1/6, 2/3, 1/2], .\
59791640643195334882500340674373891710603106494581233289572102360145176044051368853821838502221090e-1, 2^(1/3)], [[1/3, 2/3, 1/2, 1/2], .\
51915093490223523791984400933061514770956195653999304009759326094885709960576836285284594515740445e-1, -1.4263482556253197100152581984233159695007415004810879114376\
68717272602319082043789415517143646941323, (I*3^(1/2)-1)*infinity/Pi^2*3^(1/2)], [[1/6, 2/3, 1/2, 1/2], .\
26006449579947277885757863886777342782642432883082573324061397938035111196346361489658754249917948e-1, -3^(1/2)], [[2/3, 1/3, 1/2, 1/2], .\
69878518975659469241755644340510470805934107758408759496800178894892090834502989609302500785755853e-1, 2.31919053392785673153998410314004176389033052687956571052621\
0119495947514410572517316170502102292142], [[5/6, 1/3, 1/2, 1/2], .\
26006449579947277885757863886777342782642432883082573324061397938035111196346361489658754249917948e-1, 3^(1/2)], [[3/4, 1/4, 1/2, 1/2], .\
5207034476478486165140302112069151212910308695626072119106142598601078123459199688601084476616982e-2, 1.669253683348146372562859465598093617987986026980694004899654\
740207363985419052823739382320702550648], [[1/3, 1/6, 1/2, 1/2], .62063859138631951918830664834530216548277442892012316711029498246954617281970843278608631097812307\
e-1, 2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177], [[1/6, 1/6, 1/2, 1/2], .\
23762874489777959057035951948795586972510978455671573772728517399651936832572860042934271538892650e-1, 4/3*3^(1/2)], [[5/6, 5/6, 1/2, 1/2], .\
36497862071004723865798078829913646920949847208570318711671030560108254859539764740033174962323785e-1, 1/6*3^(1/2)], [[1/6, 5/6, 1/3, 1/2], .\
53863899283678419523976673712787636772988938469043961199050776090794003952860865844701224400333750e-1, -1/2*2^(2/3), (I*3^(1/2)+1)*infinity/Pi/Beta(1/2,2/3)], [[1/3
, 2/3, 1/6, 1/2], .56527740060893324860237006911898845841745249978257833146178990549234806444618792156037738363896348e-1, -2*2^(1/3), (I*3^(1/2)-1)*infinity/Pi*3^(1
/2)/Beta(1/2,5/6)], [[1/4, 3/4, 5/4, 1/2], .48985327039254602214521331767407922018675753425168589334058065474045351665391660847591827448034175e-1, 2*2^(1/2), 
undefined*I], [[3/4, 1/4, 3/4, 1/2], .69782515043065317842209831020537345636906349005988216038929698467903786864473098719380658030314078e-1, 2^(1/2)], [[1/4, 3/4, 1
/4, 1/2], .69782515043065317842209831020537345636906349005988216038929698467903786864473098719380658030314078e-1, -2^(1/2), infinity*I/Pi*2^(1/2)/Beta(1/2,3/4)], [[
1/6, 5/6, 4/3, 1/2], .17986191042409422765736956575772920959126937148353818787145975263140775119156259100676227830862672e-1, 2*2^(2/3), undefined], [[5/6, 1/6, 5/3,
1/2], .7707391324429871957806534470220359000151417737210283924146244252809373264250628961083334641717519e-2, 2*2^(1/3), 0], [[2/3, 1/3, 5/6, 3/2], .\
14172569689766397280847608448965884854255356544375581125388716245359516824578408358530776671849051e-1, 6/5*2^(2/3)], [[5/6, 1/6, 2/3, 3/2], .\
35477273886545303417034789177388628572810357171006958807825780709860428223415945721941544485531269e-1, 3/2*2^(1/3)], [[1/6, 5/6, 1/3, 3/2], .\
21509258277205051801934310900066352453539380917589990889590538701029420208805650648642433752542309e-1, 3/4*2^(2/3), (I*3^(1/2)+1)*infinity/int(2/y^(1/3)/(1-y)^(3/2)
*Pi,y = 0 .. 1,AllSolutions)], [[1/3, 2/3, 1/6, 3/2], .26404687786197627411929732378610555427472982776176503995687163406388438782627444234914038055157856e-1, 6/7*2^
(1/3), (I*3^(1/2)-1)*infinity/int(2/3/y^(1/6)/(1-y)^(3/2)*Pi*3^(1/2),y = 0 .. 1,AllSolutions)], [[1/4, 3/4, 5/4, 3/2], .\
5245788014220327934165680425889174545224995859572353958344000186558414208876771232308180741874053e-2, 8/5*2^(1/2), infinity*I/int(1/y^(5/4)/(1-y)^(3/2)*Pi*2^(1/2),y
= 0 .. 1,AllSolutions)], [[3/4, 1/4, 3/4, 3/2], .10890545764100094027725369330050343784921090856187135339428499769083657227947319691819075512676193e-1, 4/3*2^(1/2)]
, [[1/4, 3/4, 1/4, 3/2], .11675004613180594005504453787671709400657817799413954677186022174430086741243922561542250675218033e-1, 4/5*2^(1/2), infinity*I/int(1/y^(1/
4)/(1-y)^(3/2)*Pi*2^(1/2),y = 0 .. 1,AllSolutions)], [[1/6, 1/6, 0, 5/2], .\
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63557337854481162845853910464817079395188855935610111069436085492382734017279035910534743676337185e-1, 1/4*2^(1/2)], [[2/5, 2/5, 1/5, 4/5], 1.0071814687096519566600\
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59185202848712477384380363785642300263279692532687616129398699777221155001461999236306773347935102e-1, 1/4*2^(1/3)], [[1/6, 5/6, 1/2, 5/6], .\
36381826327953079356867555839609676598219050088373114289733787207371433984810080505710430378013371e-1, 1/12*3^(1/2)], [[5/3, 5/6, 1/6, 5/6], .\
83301927575911164522073451539166992003952850786589966716817725146507531765841923188749508453455542e-1, 2/3*2^(1/3), 0], [[3/2, 5/6, 1/6, 5/6], .\
31191854754932834030429133735450372714246289213022818804793453512413506722719498213783442013683764e-1, 7/24*3^(1/2), 0], [[1/7, 3/7, 5/7, 5/7], .\
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/2))^(1/3)), -1/(sin(1/7*Pi)^12+6*sin(1/7*Pi)^10*cos(1/7*Pi)^2+15*sin(1/7*Pi)^8*cos(1/7*Pi)^4+20*sin(1/7*Pi)^6*cos(1/7*Pi)^6+15*sin(1/7*Pi)^4*cos(1/7*Pi)^8+6*sin(1/
7*Pi)^2*cos(1/7*Pi)^10+cos(1/7*Pi)^12+2*sin(1/7*Pi)^6-30*sin(1/7*Pi)^4*cos(1/7*Pi)^2+30*sin(1/7*Pi)^2*cos(1/7*Pi)^4-2*cos(1/7*Pi)^6+1)^(1/2)*(-15*sin(1/7*Pi)^2*cos(
3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1
/2)+2*cos(1/7*Pi))^(1/2)-6*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*
Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2
)+2*sin(1/7*Pi)^6*cos(3/7*Pi)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-6*sin(1/7*Pi)^5*cos(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+
sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*sin(1/7*Pi)^5*sin(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+15*sin(1/7*Pi)^4*cos
(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-15*sin(1/7*Pi)^4*sin(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(
1/2)+2*cos(1/7*Pi))^(1/2)+20*sin(1/7*Pi)^3*cos(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-20*sin(1/7*Pi)^3*sin(3/7*Pi)^2*
cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-I*sin(1/7*Pi)^6*sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-
I*cos(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+
sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+2*cos(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)*sin(3/7*Pi)-I*
sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+2*I*cos(3/7*Pi
)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-12*sin(1/7*Pi)^5*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2
)+2*cos(1/7*Pi))^(1/2)-30*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+40*sin(1/7*Pi)^3*cos(3/7*
Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+30*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+
sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-12*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-20*I*sin(
1/7*Pi)^3*sin(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-15*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+
sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-6*I*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-2*I*cos(3/7*Pi)*
sin(3/7*Pi)*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-6*I*sin(1/7*Pi)^5*cos(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(
1/2)+2*cos(1/7*Pi))^(1/2)-15*I*sin(1/7*Pi)^4*cos(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*sin(1/7*Pi)^6*cos(3/7*Pi)^2*(
2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*sin(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-sin(3/7*Pi)^2*
cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-sin(1/7*Pi)^6*cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+
sin(1/7*Pi)^6*sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+cos(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7
*Pi))^(1/2)+12*I*sin(1/7*Pi)^5*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+30*I*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/
7*Pi)*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+12*I*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)
^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+20*I*sin(1/7*Pi)^3*cos(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*I*sin(1/7*Pi)^2*cos(3/7
*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*I*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+
2*cos(1/7*Pi))^(1/2)+2*I*sin(1/7*Pi)^6*cos(3/7*Pi)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+6*I*sin(1/7*Pi)^5*sin(3/7*Pi)^2*cos(1/7*
Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*I*sin(1/7*Pi)^4*sin(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))
^(1/2)-30*I*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-40*I*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*
Pi)*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2))*infinity/(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)/Beta(3/7,4/7)/Beta(2/7,6/7)], [[2/7, 3/7
, 1/7, 5/7], .40370574193883684608855191367235847749662246124715783967066907833878764472682898046925653142575613e-1, -3.20150725854689963187467469228965709496206286\
3724665104097469435548693123541410387592266366522756432], [[2/7, 4/7, 1/7, 4/7], .\
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69435548693123541410387592266366522756432], [[3/7, 4/7, 2/7, 4/7], .\
88078018281919177629921991810901591091624999711933658597095225988059258090911561419719779825830138e-1, -1/12*(-28+84*I*3^(1/2))^(1/3)-7/3/(-28+84*I*3^(1/2))^(1/3)-2
/3+1/2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3)), 1/(sin(1/7*Pi)^2+cos(1/7*Pi)^2+2*cos(1/7*Pi)+1)^(1/2)*(12*sin(1/7*Pi)^2*cos(3/7*Pi)*
sin(3/7*Pi)*cos(1/7*Pi)^2+6*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)-6*I*sin(1/7*Pi)^2*cos(3/7*Pi)^2*cos(1/7*Pi)^2-2*I*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*Pi
)-3*I*sin(1/7*Pi)^2*cos(3/7*Pi)^2*cos(1/7*Pi)-sin(1/7*Pi)^3*cos(3/7*Pi)^2+sin(1/7*Pi)^3*sin(3/7*Pi)^2-2*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)-4*sin(1/7*Pi)^3*cos(3/
7*Pi)^2*cos(1/7*Pi)+4*sin(1/7*Pi)^3*sin(3/7*Pi)^2*cos(1/7*Pi)+4*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^3-4*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^3-2*cos(3/7*Pi)*sin(
3/7*Pi)*cos(1/7*Pi)^4+3*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^2-3*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^2-2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3-I*sin(1/7*Pi)^4*
sin(3/7*Pi)^2-I*sin(3/7*Pi)^2*cos(1/7*Pi)^4-I*sin(3/7*Pi)^2*cos(1/7*Pi)^3+I*sin(1/7*Pi)^4*cos(3/7*Pi)^2+I*cos(3/7*Pi)^2*cos(1/7*Pi)^4+I*cos(3/7*Pi)^2*cos(1/7*Pi)^3+
6*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)^2+3*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)-8*I*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)+8*I*sin(1/7*Pi)*cos(
3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3+6*I*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2)*infinity/(sin(1/7*Pi)^6+3*sin(1/7*Pi)^4*cos(1/7*Pi)^2+3*sin(1/7*Pi)^2*cos(1
/7*Pi)^4+cos(1/7*Pi)^6)/Beta(3/7,4/7)/Beta(3/7,5/7)], [[1/7, 1/7, 3/7, 4/7], .\
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/3-5/2*I*3^(1/2)*(1/6*(28+84*I*3^(1/2))^(1/3)-14/3/(28+84*I*3^(1/2))^(1/3))], [[5/7, 1/7, 3/7, 4/7], .\
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9435548693123541410387592266366522756432], [[5/7, 2/7, 4/7, 4/7], .\
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, [[5/7, 1/7, 4/7, 4/7], .40240907943639400634731909714944040401663866234509128441141288245906601201135715703240860399416667e-1, 1.600753629273449815937337346144828\
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, [[4/7, 2/7, 5/7, 3/7], .40240907943639400634731909714944040401663866234509128441141288245906601201135715703240860399416667e-1, 1.600753629273449815937337346144828\
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2438448577655555996838480436431507741449], [[2/7, 1/7, 4/7, 3/7], .1221623312291496769575117849200784644012114867776589705809085692606053787111964960895582660302813\
06, -1/3*(-28+84*I*3^(1/2))^(1/3)-28/3/(-28+84*I*3^(1/2))^(1/3)+4/3-2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], [[5/7, 2/7, 3/7, 3/7]
, .20007485983747786024378727648947742559339741175107535955194851095201874277359647229619675440016996e-1, 1.70926855029938766608449599921368693939267175545535206811\
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75, 1.759488587938981416853672734385613866595459824225324761225058958693595610561327868438380921160881264], [[3/7, 1/7, 5/7, 2/7], .12359736727842401666848696287054\
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)^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], [[2/7, 1/7, 4/7, 1/7], .129921447227972874582045487520858830231885591085006542455571016624555336908588419216636583662676051
, 1.407590870351185133482938187508491093276367859380259808980047166954876488449062294750704736928705011]]:

end:



Hopefuls2:=proc():
[[[[0, 0, 0, 0], .160376328282995365630650037915397621728928601629473517662941887093415487440248087647248677678121518, 1/6*Pi^2, 1/6*Pi^2], .9142517408e-1], [[[1/2,
0, 0, 0], .215446406424598151262168999617658868512485219126761727588124913699012907249772226740948093718322976, arcsinh(15/8), 2*ln(2)], .9198665767e-1], [[[3/2, 0,
0, 0], .166836981627750005879354379653292869850893026629198799852660714262138415414482514946546100577966226, 2/3-2/3*ln(2), (-4/3+4/3*ln(2))*undefined], .9105610898\
e-1], [[[5/2, 0, 0, 0], .123120246522147760963144004235422251425266242442746382143772502329776045148689719343663790627405749, 8/5-6/5*ln(2), (-16/15+4/5*ln(2))*
undefined], .9096427979e-1], [[[5/3, 5/3, 0, 0], .108975898084621306130950358824770117731793593303480987855061282057214950716954422285424878997274086, 6.28931910992\
7904990831709560699018025197692747329836805797838909543372453576895437157774983496884236], .6854633076e-1], [[[2/3, 5/3, 0, 0], .\
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9870776246362252707767368249976424120338], .6902010054e-1], [[[4/3, 4/3, 0, 0], .13425814677987284161587612658217116568237660245803757559430401384129459869182666497\
8175406982941537, -3.392249205292772313242597822052694665355082810115925719462356169235153976407942354605156235265075211], .6907877552e-1], [[[2/3, 2/3, 0, 0], .\
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2467694061590563176941842062494106030084], .6918535314e-1], [[[4/3, 1/3, 0, 0], .\
80292413538282695578209820858592042607139681906615304682354872731858098975614075589518997038285761e-1, -.15689968211710892529703912882107786614203312404637028778494\
24676940615905631769418420624941060300844], .6889758797e-1], [[[1/3, 1/3, 0, 0], .1145078921418362632389713583308245678912832656544902974025074777029477918347222402\
60568005653457960, 1/3*Pi*3^(1/2)], .6994292759e-1], [[[1/3, 2/3, 1/2, 0], .\
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0980614427148972704944106278399727537177, 1/6*Pi^(3/2)*(I*3^(1/2)-1)/GAMMA(2/3)/GAMMA(5/6)*3^(1/2)], .1545305638e-2], [[[2/3, 1/3, 1/2, 0], .\
8512659681647715474312113899794985919242782480448555885110711696075396491135209415458676916925656e-2, 1.293554779614895267476757512565605818828929257420200788177626\
770763033747278022225220322143942310013, -1/4/Pi^(1/2)*(3*I+3^(1/2))*GAMMA(2/3)*GAMMA(5/6)*3^(1/2)], .8813140347e-3], [[[1/3, 2/3, 3/2, 0], .\
18870440581602688763698192575873349249798362219652008617619083406016692194631613644919279551255572e-1, -.70109105266272713058750953952514706773151110271199304809099\
69935381423829909016480354261332425123924, 0], .1315446466e-2], [[[2/3, 1/3, 3/2, 0], .\
11523429780165650732046049820208170280048644329061030500497508618642172149798877315665719267672350e-1, .431184926538298422492252504188535272942976419140066929392542\
2569210112490926740750734407146474366711, 0], .2944543401e-3], [[[1/3, 2/3, 5/2, 0], .\
12178173924175540179499094164125184835472752517283072929116754854283575503928909719683401277717602e-1, -1.6358791228796966380375222588920098247068592396613171122123\
26318255665560312103845415994310899195582, 0], .7790353411e-3], [[[1/3, 0, 5/3, 0], .\
64102958178967597786497191815391791080609952668713253490321907401019356059545234156874039416987197e-1, -.10459978807807261686469275254738524409468874936424685852329\
49784627077270421179612280416627373533896, undefined], .6889035589e-1], [[[1/2, 1/2, 5/3, 0], .\
52122665489671418843991262359262154823048120605590975603982469306691909746901768423497191156444029e-1, -.70109105266272713058750953952514706773151110271199304809099\
69935381423829909016480354261332425123924, 0], .2222391869e-2], [[[2/3, 0, 4/3, 0], .\
77440410652469971117584683804280108413696392053447793078410426809700374570167977158676255031614567e-1, .802299894039036308432346376273692622047344374682123429261647\
4892313538635210589806140208313686766948, undefined], .6914340679e-1], [[[1/2, 1/2, 4/3, 0], .\
12856811824821736978725300168073003933869095642604059788924795180518596409583120933356246521283485e-1, .431184926538298422492252504188535272942976419140066929392542\
2569210112490926740750734407146474366711, 0], .6842761305e-3], [[[1/3, 0, 2/3, 0], .11427862546130078664359706862777091394508673195933009972088574673925513900968004\
1134955153344236286, 2/9*Pi*3^(1/2), 2/9*Pi*3^(1/2)], .6988065559e-1], [[[1/2, 1/2, 2/3, 0], .\
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9027684044996370696300293762858589746685, 2*GAMMA(2/3)/Pi^(1/2)*GAMMA(5/6)], .9112684198e-3], [[[2/3, 1/3, 5/6, 1/2], .\
50650410384634165913813072620628502426328782155099083611022322798493659258463300786019388908401531e-1, 2^(2/3)], .4874301014e-1], [[[5/6, 1/6, 2/3, 1/2], .\
59791640643195334882500340674373891710603106494581233289572102360145176044051368853821838502221090e-1, 2^(1/3)], .4816610407e-1], [[[1/3, 2/3, 1/2, 1/2], .\
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23762874489777959057035951948795586972510978455671573772728517399651936832572860042934271538892650e-1, 4/3*3^(1/2)], .2341824205e-1], [[[5/6, 5/6, 1/2, 1/2], .\
36497862071004723865798078829913646920949847208570318711671030560108254859539764740033174962323785e-1, 1/6*3^(1/2)], .2345436372e-1], [[[1/6, 5/6, 1/3, 1/2], .\
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48985327039254602214521331767407922018675753425168589334058065474045351665391660847591827448034175e-1, 2*2^(1/2), undefined*I], .2938186868e-1], [[[3/4, 1/4, 3/4, 1
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5245788014220327934165680425889174545224995859572353958344000186558414208876771232308180741874053e-2, 8/5*2^(1/2), infinity*I/int(1/y^(5/4)/(1-y)^(3/2)*Pi*2^(1/2),y
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10890545764100094027725369330050343784921090856187135339428499769083657227947319691819075512676193e-1, 4/3*2^(1/2)], .2866393517e-1], [[[1/4, 3/4, 1/4, 3/2], .\
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28314611737932654785998523269163299017914491162617131585632257982332946008293451120913102946660449e-1, 7/27*2^(2/3)], .4677451012e-1], [[[5/6, 1/6, 2/3, 5/3], .\
14172569689766397280847608448965884854255356544375581125388716245359516824578408358530776671849051e-1, 6/5*2^(2/3)], .4835506878e-1], [[[1/3, 2/3, 1/6, 5/3], .\
21509258277205051801934310900066352453539380917589990889590538701029420208805650648642433752542309e-1, 3/4*2^(2/3)], .4864507296e-1], [[[1/3, 2/3, 5/6, 5/3], .\
37690357222342528523978113919225409452120273064285035908071846104520044489760302841191059022212229e-1, 7/24*2^(2/3)], .4690823866e-1], [[[2/3, 1/3, 5/6, 4/3], .\
35477273886545303417034789177388628572810357171006958807825780709860428223415945721941544485531269e-1, 3/2*2^(1/3)], .4729332713e-1], [[[1/6, 5/6, 1/3, 4/3], .\
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21118643847597301842442022962699612658917563609447119625779468729386445069119853529319055329962545e-1, 27/14*2^(1/3), undefined], .4734452372e-1], [[[1/6, 5/6, 3/2,
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53161134223179944882842740674694108209193507949530637443516134026460493859109825088357764715603835e-1, 1/4*2^(2/3)], .4783042545e-1], [[[1/6, 5/6, 2/3, 2/3], .\
60797634440080760374877456949148357252327855355538251568294856421404426118128942833387568756337013e-1, 1/6*2^(2/3)], .4777104453e-1], [[[5/6, 1/6, 2/3, 2/3], .\
50650410384634165913813072620628502426328782155099083611022322798493659258463300786019388908401531e-1, 2^(2/3)], .4872535023e-1], [[[5/6, 1/6, 1/2, 2/3], .\
26006449579947277885757863886777342782642432883082573324061397938035111196346361489658754249917948e-1, 3^(1/2)], .2299192949e-1], [[[5/6, 5/6, 1/3, 2/3], .\
56049434604400065697488328396559516382689443106555333392984209747945599169464643140568397071012506e-1, 1/3*2^(1/3)], .4868037152e-1], [[[1/6, 2/3, 1/3, 2/3], .\
53863899283678419523976673712787636772988938469043961199050776090794003952860865844701224400333750e-1, -1/2*2^(2/3)], .4885750238e-1], [[[1/6, 5/6, 5/3, 2/3], .\
7226053080472746235938532833531782254025196073520483322946400095352182963221490071688154192201137e-2, 7/12*2^(2/3), 0], .4644601775e-1], [[[5/6, 1/6, 5/3, 2/3], .\
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47161319885690265544397908540337481729759025538971811611496175673761553057171842921996528476935652e-1, 2^(1/3)], .4866808045e-1], [[[1/6, 5/6, 1/3, 1/3], .\
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72462517832845913459157745715244683282063611734719405007398551604232426039510686275867092509168135e-1, 3/4*2^(2/3)], .4853731578e-1], [[[4/3, 1/6, 5/6, 1/6], .\
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59185202848712477384380363785642300263279692532687616129398699777221155001461999236306773347935102e-1, 1/4*2^(1/3)], .4876856501e-1], [[[1/6, 5/6, 1/2, 5/6], .\
36381826327953079356867555839609676598219050088373114289733787207371433984810080505710430378013371e-1, 1/12*3^(1/2)], .2343757002e-1], [[[5/3, 5/6, 1/6, 5/6], .\
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/3+1/2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], .5089984286e-1], [[[1/7, 4/7, 2/7, 5/7], .\
68965186249222348994391517892275275357479886234695397781877723401193299450525728434921941634659305e-1, -1/12*(-28+84*I*3^(1/2))^(1/3)-7/3/(-28+84*I*3^(1/2))^(1/3)+1
/3+1/2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], .4575921189e-1], [[[4/7, 3/7, 1/7, 5/7], .105566623617240268384579783013748426622014\
567541413755923329382884591000074478843194264508981329730, -1/12*(28+84*I*3^(1/2))^(1/3)-7/3/(28+84*I*3^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/6*(28+84*I*3^(1/2))^(1/3)-\
14/3/(28+84*I*3^(1/2))^(1/3)), -1/(sin(1/7*Pi)^12+6*sin(1/7*Pi)^10*cos(1/7*Pi)^2+15*sin(1/7*Pi)^8*cos(1/7*Pi)^4+20*sin(1/7*Pi)^6*cos(1/7*Pi)^6+15*sin(1/7*Pi)^4*cos(
1/7*Pi)^8+6*sin(1/7*Pi)^2*cos(1/7*Pi)^10+cos(1/7*Pi)^12+2*sin(1/7*Pi)^6-30*sin(1/7*Pi)^4*cos(1/7*Pi)^2+30*sin(1/7*Pi)^2*cos(1/7*Pi)^4-2*cos(1/7*Pi)^6+1)^(1/2)*(-15*
sin(1/7*Pi)^2*cos(3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2
+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-6*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*sin(1/7*Pi)*sin(3
/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*
cos(1/7*Pi))^(1/2)-I*sin(1/7*Pi)^6*sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-I*cos(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7
*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*sin(1/7*Pi)^6*cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*sin(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1
/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+2*sin(1/7*Pi)^6*cos(3/7*Pi)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-6*sin(1/7*Pi)
^5*cos(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*sin(1/7*Pi)^5*sin(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^
(1/2)-2*cos(1/7*Pi))^(1/2)+15*sin(1/7*Pi)^4*cos(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-15*sin(1/7*Pi)^4*sin(3/7*Pi)^2*
cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+20*sin(1/7*Pi)^3*cos(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(
1/7*Pi))^(1/2)-20*sin(1/7*Pi)^3*sin(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^
2)^(1/2)+2*cos(1/7*Pi))^(1/2)+sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+2*cos(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1
/7*Pi))^(1/2)*sin(3/7*Pi)-I*sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+I*cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1
/7*Pi))^(1/2)+2*I*cos(3/7*Pi)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-12*sin(1/7*Pi)^5*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)*(2*(cos(1
/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-30*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2
)+40*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+30*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/
7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-12*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*
cos(1/7*Pi))^(1/2)-20*I*sin(1/7*Pi)^3*sin(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-15*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1
/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-6*I*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi
))^(1/2)-2*I*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-6*I*sin(1/7*Pi)^5*cos(3/7*Pi)^2*cos(1/7*Pi)*(2*(cos(1
/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-15*I*sin(1/7*Pi)^4*cos(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)-sin(1/
7*Pi)^6*cos(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+sin(1/7*Pi)^6*sin(3/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^
(1/2)+cos(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-sin(3/7*Pi)^2*cos(1/7*Pi)^6*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*
cos(1/7*Pi))^(1/2)+12*I*sin(1/7*Pi)^5*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+30*I*sin(1/7*Pi)^2*cos(3/7*Pi)
*sin(3/7*Pi)*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+12*I*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(
1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+20*I*sin(1/7*Pi)^3*cos(3/7*Pi)^2*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*I*sin(1/7*Pi)^2*
cos(3/7*Pi)^2*cos(1/7*Pi)^4*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2)+6*I*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^5*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)
^(1/2)+2*cos(1/7*Pi))^(1/2)+2*I*sin(1/7*Pi)^6*cos(3/7*Pi)*sin(3/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+6*I*sin(1/7*Pi)^5*sin(3/7*Pi)^2*
cos(1/7*Pi)*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)+15*I*sin(1/7*Pi)^4*sin(3/7*Pi)^2*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(
1/7*Pi))^(1/2)-30*I*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)+2*cos(1/7*Pi))^(1/2)-40*I*sin(1/7*Pi)^3*cos(3/7*Pi)*
sin(3/7*Pi)*cos(1/7*Pi)^3*(2*(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)-2*cos(1/7*Pi))^(1/2))*infinity/(cos(1/7*Pi)^2+sin(1/7*Pi)^2)^(1/2)/Beta(3/7,4/7)/Beta(2/7,6/7)], .\
5653944712e-1], [[[2/7, 3/7, 1/7, 5/7], .40370574193883684608855191367235847749662246124715783967066907833878764472682898046925653142575613e-1, -3.20150725854689963\
1874674692289657094962062863724665104097469435548693123541410387592266366522756432], .2424631612e-1], [[[2/7, 4/7, 1/7, 4/7], .\
40370574193883684608855191367235847749662246124715783967066907833878764472682898046925653142575613e-1, -3.2015072585468996318746746922896570949620628637246651040974\
69435548693123541410387592266366522756432], .2425636574e-1], [[[3/7, 4/7, 2/7, 4/7], .\
88078018281919177629921991810901591091624999711933658597095225988059258090911561419719779825830138e-1, -1/12*(-28+84*I*3^(1/2))^(1/3)-7/3/(-28+84*I*3^(1/2))^(1/3)-2
/3+1/2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3)), 1/(sin(1/7*Pi)^2+cos(1/7*Pi)^2+2*cos(1/7*Pi)+1)^(1/2)*(-sin(1/7*Pi)^3*cos(3/7*Pi)^2+
sin(1/7*Pi)^3*sin(3/7*Pi)^2+12*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2+6*sin(1/7*Pi)^2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)-6*I*sin(1/7*Pi)^2*cos(3/7*Pi)
^2*cos(1/7*Pi)^2-2*I*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*Pi)-3*I*sin(1/7*Pi)^2*cos(3/7*Pi)^2*cos(1/7*Pi)-2*sin(1/7*Pi)^4*cos(3/7*Pi)*sin(3/7*Pi)-4*sin(1/7*Pi)^3*cos(3
/7*Pi)^2*cos(1/7*Pi)+4*sin(1/7*Pi)^3*sin(3/7*Pi)^2*cos(1/7*Pi)+4*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^3-4*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^3-2*cos(3/7*Pi)*sin
(3/7*Pi)*cos(1/7*Pi)^4+3*sin(1/7*Pi)*cos(3/7*Pi)^2*cos(1/7*Pi)^2-3*sin(1/7*Pi)*sin(3/7*Pi)^2*cos(1/7*Pi)^2-2*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3-I*sin(1/7*Pi)^4*
sin(3/7*Pi)^2-I*sin(3/7*Pi)^2*cos(1/7*Pi)^4-I*sin(3/7*Pi)^2*cos(1/7*Pi)^3+I*sin(1/7*Pi)^4*cos(3/7*Pi)^2+I*cos(3/7*Pi)^2*cos(1/7*Pi)^4+I*cos(3/7*Pi)^2*cos(1/7*Pi)^3-\
8*I*sin(1/7*Pi)^3*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)+6*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)^2+3*I*sin(1/7*Pi)^2*sin(3/7*Pi)^2*cos(1/7*Pi)+8*I*sin(1/7*Pi)*cos(
3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^3+6*I*sin(1/7*Pi)*cos(3/7*Pi)*sin(3/7*Pi)*cos(1/7*Pi)^2)*infinity/(sin(1/7*Pi)^6+3*sin(1/7*Pi)^4*cos(1/7*Pi)^2+3*sin(1/7*Pi)^2*cos(1
/7*Pi)^4+cos(1/7*Pi)^6)/Beta(3/7,4/7)/Beta(3/7,5/7)], .5091046478e-1], [[[1/7, 1/7, 3/7, 4/7], .\
81540603130989681635904789036432040951594683791108706496858621582945088816543450737743235457288671e-1, -5/12*(28+84*I*3^(1/2))^(1/3)-35/3/(28+84*I*3^(1/2))^(1/3)+10
/3-5/2*I*3^(1/2)*(1/6*(28+84*I*3^(1/2))^(1/3)-14/3/(28+84*I*3^(1/2))^(1/3))], .5646093677e-1], [[[3/7, 2/7, 4/7, 4/7], .\
40370574193883684608855191367235847749662246124715783967066907833878764472682898046925653142575613e-1, 3.20150725854689963187467469228965709496206286372466510409746\
9435548693123541410387592266366522756432], .2424356552e-1], [[[5/7, 2/7, 4/7, 4/7], .\
88078018281919177629921991810901591091624999711933658597095225988059258090911561419719779825830138e-1, 1/6*(28+84*I*3^(1/2))^(1/3)+14/3/(28+84*I*3^(1/2))^(1/3)+2/3]
, .5086926078e-1], [[[5/7, 1/7, 4/7, 4/7], .40240907943639400634731909714944040401663866234509128441141288245906601201135715703240860399416667e-1, 1.600753629273449\
815937337346144828547481031431862332552048734717774346561770705193796133183261378216], .2420566272e-1], [[[3/7, 1/7, 5/7, 4/7], .\
22484766844264544479390930223782163050452172118954328541375955582593363323072508956791302734671867e-1, 1.83537817448657947703172958916641892025017393557074950568053\
5317940230809527228085264994118764699311], .2650774072e-1], [[[4/7, 3/7, 5/7, 3/7], .\
88078018281919177629921991810901591091624999711933658597095225988059258090911561419719779825830138e-1, 1/6*(28+84*I*3^(1/2))^(1/3)+14/3/(28+84*I*3^(1/2))^(1/3)+2/3]
, .5087560228e-1], [[[4/7, 2/7, 5/7, 3/7], .40240907943639400634731909714944040401663866234509128441141288245906601201135715703240860399416667e-1, 1.600753629273449\
815937337346144828547481031431862332552048734717774346561770705193796133183261378216], .2421579677e-1], [[[2/7, 1/7, 5/7, 3/7], .\
55210307436119093878591161105780910246949329956504197093145473725407787883487155564595538949084380e-1, 1.58218866817255634146036173069600995441011038342026842581914\
2438448577655555996838480436431507741449], .2933164798e-1], [[[2/7, 1/7, 4/7, 3/7], .1221623312291496769575117849200784644012114867776589705809085692606053787111964\
96089558266030281306, -1/3*(-28+84*I*3^(1/2))^(1/3)-28/3/(-28+84*I*3^(1/2))^(1/3)+4/3-2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], .\
5082602393e-1], [[[5/7, 2/7, 1/7, 2/7], .49930196017904037493722134677636770309684950974266965570445891366683105653869070714678343914623318e-1, 1/3*(-28+84*I*3^(1/2
))^(1/3)+28/3/(-28+84*I*3^(1/2))^(1/3)-4/3], .4572847263e-1], [[[4/7, 2/7, 1/7, 2/7], .\
86437813845228667397080863352785443661973608971473755243597893818197917736766023182394602195496222e-1, 1.83259567068458360354172661276544118118565846415490113859512\
0535382678993805458866779467188498922960], .2255786762e-1], [[[5/7, 1/7, 2/7, 2/7], .\
86317887995754242118846051821596241440753907102595370601850876885566124103650030389204040067762268e-1, 1.37444675301343770265629495957408088588924384811617585394634\
0401537009245354094150084600391374192220], .2252700143e-1], [[[1/7, 1/7, 3/7, 2/7], .1203344614191030396294638538527480880202968182400648988516951034098080502681368\
96776254288566090775, 1.759488587938981416853672734385613866595459824225324761225058958693595610561327868438380921160881264], .2413871059e-1], [[[3/7, 1/7, 5/7, 2/7
], .123597367278424016668486962870541019852310230020915237766487230677250205619075282699225659139151051, -1/4*(-28+84*I*3^(1/2))^(1/3)-7/(-28+84*I*3^(1/2))^(1/3)+1-\
3/2*I*3^(1/2)*(1/6*(-28+84*I*3^(1/2))^(1/3)-14/3/(-28+84*I*3^(1/2))^(1/3))], .5081540372e-1], [[[2/7, 1/7, 4/7, 1/7], .129921447227972874582045487520858830231885591\
085006542455571016624555336908588419216636583662676051, 1.407590870351185133482938187508491093276367859380259808980047166954876488449062294750704736928705011], .\
2411645952e-1]]:
end:




#Hopefuls1C(r): A list of length 21 of equivalence classes of qudraples based on Hopefuls1. Try:
#Hopefuls1C(r);
Hopefuls1C:=proc(r):
[[[[0, 0, 0, 0], r[1]]], [[[0, 2/3, 5/6, 5/3], r[2]], [[2/3, 2/3, 1/2, 0], 11/6-2/3*r[2]], [[2/3, 2/3, 3/2, 0], -1/6-2/3*r[2]], [[2/3, 2/3, 5/2, 0], -9/14-10/7*r[2]
], [[2/3, 5/3, 1/2, 0], 23/6-5/3*r[2]], [[2/3, 5/3, 5/2, 0], 9/2+5*r[2]], [[5/2, 5/6, 2/3, 5/6], 17/16+5/8*r[2]], [[5/3, 2/3, 3/2, 0], -8/3+4/3*r[2]], [[5/3, 2/3, 5
/2, 0], -4*r[2]], [[5/3, 5/3, 1/2, 0], 289/84-55/42*r[2]], [[5/3, 5/3, 3/2, 0], -11/12+5/6*r[2]], [[5/6, 1/2, 2/3, 0], 11/9-4/9*r[2]], [[5/6, 2/3, 0, 2/3], 1/6+2/3*
r[2]], [[5/6, 3/2, 2/3, 0], 16/9-8/9*r[2]], [[5/6, 5/2, 2/3, 0], 2-10/9*r[2]]], [[[1/2, 0, 0, 0], r[3]], [[3/2, 0, 0, 0], 2/3-1/3*r[3]], [[5/2, 0, 0, 0], 8/5-3/5*r[
3]]], [[[1/2, 1/2, 1/3, 0], r[4]], [[1/2, 1/2, 4/3, 0], 1/4*r[4]], [[2/3, 1/3, 1/2, 0], 3/4*r[4]], [[2/3, 1/3, 1/2, 1/2], 4/r[4]], [[2/3, 1/3, 3/2, 0], 1/4*r[4]]],
[[[1/2, 1/2, 2/3, 0], r[5]], [[1/2, 1/2, 5/3, 0], -1/2*r[5]], [[1/2, 1/6, 2/3, 1/3], r[5]], [[1/3, 1/6, 1/2, 1/2], 3/2*r[5]], [[1/3, 2/3, 1/2, 0], 3/2*r[5]], [[1/3,
2/3, 1/2, 1/2], -2/r[5]], [[1/3, 2/3, 3/2, 0], -1/2*r[5]], [[1/3, 2/3, 5/2, 0], -7/6*r[5]]], [[[1/2, 1/6, 5/6, 1/6], r[6]], [[1/6, 1/6, 1/2, 1/2], 2*r[6]], [[1/6, 2
/3, 1/2, 1/2], -2/r[6]], [[1/6, 5/6, 1/2, 1/3], -2/r[6]], [[1/6, 5/6, 1/2, 5/6], 1/6/r[6]], [[1/6, 5/6, 3/2, 1/3], 6*r[6]], [[1/6, 5/6, 3/2, 4/3], 32/7/r[6]], [[1/6
, 5/6, 5/2, 1/3], 12*r[6]], [[3/2, 1/6, 5/6, 1/6], 1/2*r[6]], [[3/2, 5/6, 1/6, 5/6], 7/12/r[6]], [[5/2, 1/6, 5/6, 1/6], -(5+5*r[6])/(-8-6*r[6])], [[5/6, 1/3, 1/2, 1
/2], 2/r[6]], [[5/6, 1/6, 1/2, 2/3], 2/r[6]], [[5/6, 5/6, 1/2, 1/2], 1/4*r[6]]], [[[1/3, 0, 2/3, 0], r[7]], [[1/3, 0, 5/3, 0], 1/2-1/2*r[7]], [[1/3, 1/3, 0, 0], 3/2
*r[7]], [[2/3, 0, 4/3, 0], 1/2+1/4*r[7]], [[2/3, 2/3, 0, 0], 3/2+3/4*r[7]], [[2/3, 5/3, 0, 0], 9/2+3*r[7]], [[4/3, 1/3, 0, 0], 3/4-3/4*r[7]], [[4/3, 4/3, 0, 0], -9/
8-15/8*r[7]], [[5/3, 5/3, 0, 0], 15/4+21/10*r[7]]], [[[1/3, 1/3, 1/2, 0], r[8]], [[1/3, 1/3, 3/2, 0], 2/5-1/5*r[8]], [[1/3, 1/3, 5/2, 0], 52/55-21/55*r[8]], [[1/3,
1/6, 1/2, 1/6], r[8]], [[1/3, 4/3, 1/2, 0], -3/2-1/2*r[8]], [[1/3, 4/3, 3/2, 0], 5/2-1/2*r[8]], [[1/3, 4/3, 5/2, 0], 97/10-21/10*r[8]], [[1/6, 1/2, 1/3, 0], 4/3*r[8
]], [[1/6, 1/2, 4/3, 0], 1/2-1/6*r[8]], [[1/6, 1/3, 0, 1/3], 2*r[8]], [[1/6, 1/3, 0, 4/3], -3/4-1/4*r[8]], [[1/6, 1/6, 0, 1/2], 2*r[8]], [[1/6, 1/6, 0, 5/2], -14/5-\
12/5*r[8]], [[1/6, 1/6, 1/3, 1/3], 4/3*r[8]], [[1/6, 1/6, 4/3, 4/3], -1/3-r[8]], [[1/6, 3/2, 4/3, 0], 5/3-1/3*r[8]], [[1/6, 4/3, 0, 4/3], 7/20+1/20*r[8]], [[1/6, 5/
2, 1/3, 0], -28/9-8/3*r[8]], [[1/6, 5/2, 4/3, 0], 19/18-1/6*r[8]], [[4/3, 1/3, 1/2, 0], 3/5-1/5*r[8]], [[4/3, 1/3, 5/2, 0], -1/5+3/5*r[8]], [[4/3, 4/3, 1/2, 0], -9/
20-7/20*r[8]], [[4/3, 4/3, 3/2, 0], 7/4+1/4*r[8]], [[4/3, 4/3, 5/2, 0], 19/4-3/4*r[8]], [[5/2, 1/6, 4/3, 1/6], 1/4+3/4*r[8]]], [[[1/3, 1/6, 5/6, 1/6], r[9]], [[1/3,
2/3, 1/6, 1/2], -3/r[9]], [[1/3, 2/3, 1/6, 3/2], 9/7/r[9]], [[1/3, 2/3, 1/6, 5/3], r[9]], [[1/3, 2/3, 5/6, 2/3], 1/3*r[9]], [[1/3, 2/3, 5/6, 5/3], 7/18*r[9]], [[1/6
, 1/3, 2/3, 1/3], 9/4/r[9]], [[1/6, 1/6, 2/3, 1/3], 4/3*r[9]], [[1/6, 2/3, 1/3, 2/3], -2/3*r[9]], [[1/6, 5/6, 1/3, 1/2], -2/3*r[9]], [[1/6, 5/6, 1/3, 1/3], -3/r[9]]
, [[1/6, 5/6, 1/3, 3/2], r[9]], [[1/6, 5/6, 1/3, 4/3], 9/7/r[9]], [[1/6, 5/6, 2/3, 2/3], 2/9*r[9]], [[1/6, 5/6, 2/3, 5/3], 28/81*r[9]], [[1/6, 5/6, 2/3, 5/6], 3/8/r
[9]], [[1/6, 5/6, 4/3, 1/2], 8/3*r[9]], [[1/6, 5/6, 4/3, 1/3], 9/2/r[9]], [[1/6, 5/6, 4/3, 4/3], 81/28/r[9]], [[1/6, 5/6, 5/3, 2/3], 7/9*r[9]], [[2/3, 1/3, 1/6, 4/3
], 3/8/r[9]], [[2/3, 1/3, 5/6, 1/2], 4/3*r[9]], [[2/3, 1/3, 5/6, 3/2], 8/5*r[9]], [[2/3, 1/3, 5/6, 4/3], 9/4/r[9]], [[4/3, 1/6, 5/6, 1/6], 2/3*r[9]], [[5/3, 5/6, 1/
6, 5/6], 1/r[9]], [[5/6, 1/3, 2/3, 1/3], 3/2/r[9]], [[5/6, 1/6, 1/3, 1/3], 3/2/r[9]], [[5/6, 1/6, 1/3, 4/3], 1/2/r[9]], [[5/6, 1/6, 2/3, 1/2], 3/2/r[9]], [[5/6, 1/6
, 2/3, 2/3], 4/3*r[9]], [[5/6, 1/6, 2/3, 3/2], 9/4/r[9]], [[5/6, 1/6, 2/3, 5/3], 8/5*r[9]], [[5/6, 1/6, 4/3, 1/3], 3/4/r[9]], [[5/6, 1/6, 4/3, 4/3], 5/8/r[9]], [[5/
6, 1/6, 5/3, 1/2], 3/r[9]], [[5/6, 1/6, 5/3, 2/3], 2*r[9]], [[5/6, 1/6, 5/3, 5/3], 9/5*r[9]], [[5/6, 5/6, 1/3, 2/3], 1/2/r[9]]], [[[1/4, 1/4, 3/4, 1/4], r[10]], [[1
/4, 3/4, 1/4, 1/2], -r[10]], [[1/4, 3/4, 1/4, 3/2], 4/5*r[10]], [[1/4, 3/4, 3/4, 3/4], 1/2/r[10]], [[1/4, 3/4, 5/4, 1/2], 2*r[10]], [[1/4, 3/4, 5/4, 3/2], -(16+8*r[
10])/(-5-5*r[10])], [[1/4, 5/4, 3/4, 1/4], 1/2/r[10]], [[3/4, 1/4, 3/4, 1/2], r[10]], [[3/4, 1/4, 3/4, 3/2], 8/3/r[10]], [[3/4, 1/4, 5/4, 1/4], 1/r[10]], [[5/4, 5/4
, 3/4, 1/4], 3/4/r[10]]], [[[1/4, 3/4, 3/2, 0], r[11]]], [[[1/5, 1/5, 3/5, 2/5], r[12]], [[1/5, 3/5, 2/5, 3/5], -r[12]/(3-r[12])], [[2/5, 1/5, 4/5, 1/5], 2/3*r[12]]
, [[2/5, 2/5, 1/5, 4/5], -r[12]/(3-r[12])], [[2/5, 3/5, 4/5, 4/5], -1/2*(-3+r[12])/r[12]], [[3/5, 4/5, 1/5, 4/5], -(1-r[12])/(6-r[12])], [[4/5, 1/5, 3/5, 3/5], -r[
12]/(-3+r[12])], [[4/5, 2/5, 3/5, 2/5], -r[12]/(-3+r[12])]], [[[1/7, 1/7, 3/7, 2/7], r[13]], [[2/7, 1/7, 4/7, 1/7], 4/5*r[13]]], [[[1/7, 1/7, 3/7, 4/7], r[14]], [[1
/7, 4/7, 2/7, 5/7], -r[14]/(5-r[14])], [[2/7, 1/7, 4/7, 3/7], 4-4/5*r[14]], [[3/7, 1/7, 5/7, 2/7], 3-3/5*r[14]], [[3/7, 3/7, 2/7, 5/7], -5/(5-r[14])], [[3/7, 4/7, 2
/7, 4/7], -5/(5-r[14])], [[4/7, 3/7, 1/7, 5/7], -5/r[14]], [[4/7, 3/7, 5/7, 3/7], -5/(-5+r[14])], [[5/7, 2/7, 1/7, 2/7], -(-10+2*r[14])/r[14]], [[5/7, 2/7, 4/7, 4/7
], -5/(-5+r[14])]], [[[1/7, 3/7, 5/7, 5/7], r[15]], [[2/7, 1/7, 5/7, 3/7], 1-1/3*r[15]]], [[[2/7, 3/7, 1/7, 5/7], r[16]], [[2/7, 4/7, 1/7, 4/7], r[16]], [[3/7, 2/7,
4/7, 4/7], -r[16]], [[4/7, 1/7, 3/7, 5/7], -r[16]], [[4/7, 2/7, 5/7, 3/7], -1/2*r[16]], [[5/7, 1/7, 4/7, 4/7], -1/2*r[16]]], [[[2/7, 4/7, 3/7, 5/7], r[17]], [[3/7,
3/7, 4/7, 5/7], 2*r[17]]], [[[3/4, 1/4, 1/2, 1/2], r[18]]], [[[3/7, 1/7, 5/7, 4/7], r[19]]], [[[4/7, 2/7, 1/7, 2/7], r[20]], [[5/7, 1/7, 2/7, 2/7], 3/4*r[20]]], [[[
5/7, 1/7, 3/7, 4/7], r[21]], [[5/7, 2/7, 3/7, 3/7], r[21]]]]:
end:


#Hopefuls2C(r): A list of length 16 of equivalence classes of qudruples based on Hopefuls2(). Try:
#Hopefuls2C(r);

Hopefuls2C:=proc(r):
[[[[0, 0, 0, 0], r[1]]], [[[1/2, 0, 0, 0], r[2]], [[3/2, 0, 0, 0], 2/3-1/3*r[2]], [[5/2, 0, 0, 0], 8/5-3/5*r[2]]], [[[1/2, 1/2, 1/3, 0], r[3]], [[1/2, 1/2, 4/3, 0],
1/4*r[3]], [[2/3, 1/3, 1/2, 0], 3/4*r[3]], [[2/3, 1/3, 1/2, 1/2], 4/r[3]], [[2/3, 1/3, 3/2, 0], 1/4*r[3]]], [[[1/2, 1/2, 2/3, 0], r[4]], [[1/2, 1/2, 5/3, 0], -1/2*r
[4]], [[1/2, 1/6, 2/3, 1/3], r[4]], [[1/3, 1/6, 1/2, 1/2], 3/2*r[4]], [[1/3, 2/3, 1/2, 0], 3/2*r[4]], [[1/3, 2/3, 1/2, 1/2], -2/r[4]], [[1/3, 2/3, 3/2, 0], -1/2*r[4
]], [[1/3, 2/3, 5/2, 0], -7/6*r[4]]], [[[1/2, 1/6, 5/6, 1/6], r[5]], [[1/6, 1/6, 1/2, 1/2], 2*r[5]], [[1/6, 2/3, 1/2, 1/2], -2/r[5]], [[1/6, 5/6, 1/2, 1/3], -2/r[5]
], [[1/6, 5/6, 1/2, 5/6], 1/6/r[5]], [[1/6, 5/6, 3/2, 1/3], 6*r[5]], [[1/6, 5/6, 3/2, 4/3], 32/7/r[5]], [[1/6, 5/6, 5/2, 1/3], 12*r[5]], [[3/2, 1/6, 5/6, 1/6], 1/2*
r[5]], [[3/2, 5/6, 1/6, 5/6], 7/12/r[5]], [[5/2, 1/6, 5/6, 1/6], -(5+5*r[5])/(-8-6*r[5])], [[5/6, 1/3, 1/2, 1/2], 2/r[5]], [[5/6, 1/6, 1/2, 2/3], 2/r[5]], [[5/6, 5/
6, 1/2, 1/2], 1/4*r[5]]], [[[1/3, 0, 2/3, 0], r[6]], [[1/3, 0, 5/3, 0], 1/2-1/2*r[6]], [[1/3, 1/3, 0, 0], 3/2*r[6]], [[2/3, 0, 4/3, 0], 1/2+1/4*r[6]], [[2/3, 2/3, 0
, 0], 3/2+3/4*r[6]], [[2/3, 5/3, 0, 0], 9/2+3*r[6]], [[4/3, 1/3, 0, 0], 3/4-3/4*r[6]], [[4/3, 4/3, 0, 0], -9/8-15/8*r[6]], [[5/3, 5/3, 0, 0], 15/4+21/10*r[6]]], [[[
1/3, 1/6, 5/6, 1/6], r[7]], [[1/3, 2/3, 1/6, 1/2], -3/r[7]], [[1/3, 2/3, 1/6, 3/2], 9/7/r[7]], [[1/3, 2/3, 1/6, 5/3], r[7]], [[1/3, 2/3, 5/6, 2/3], 1/3*r[7]], [[1/3
, 2/3, 5/6, 5/3], 7/18*r[7]], [[1/6, 1/3, 2/3, 1/3], 9/4/r[7]], [[1/6, 1/6, 2/3, 1/3], 4/3*r[7]], [[1/6, 2/3, 1/3, 2/3], -2/3*r[7]], [[1/6, 5/6, 1/3, 1/2], -2/3*r[7
]], [[1/6, 5/6, 1/3, 1/3], -3/r[7]], [[1/6, 5/6, 1/3, 3/2], r[7]], [[1/6, 5/6, 1/3, 4/3], 9/7/r[7]], [[1/6, 5/6, 2/3, 2/3], 2/9*r[7]], [[1/6, 5/6, 2/3, 5/3], 28/81*
r[7]], [[1/6, 5/6, 2/3, 5/6], 3/8/r[7]], [[1/6, 5/6, 4/3, 1/2], 8/3*r[7]], [[1/6, 5/6, 4/3, 1/3], 9/2/r[7]], [[1/6, 5/6, 4/3, 4/3], 81/28/r[7]], [[1/6, 5/6, 5/3, 2/
3], 7/9*r[7]], [[2/3, 1/3, 1/6, 4/3], 3/8/r[7]], [[2/3, 1/3, 5/6, 1/2], 4/3*r[7]], [[2/3, 1/3, 5/6, 3/2], 8/5*r[7]], [[2/3, 1/3, 5/6, 4/3], 9/4/r[7]], [[4/3, 1/6, 5
/6, 1/6], 2/3*r[7]], [[5/3, 5/6, 1/6, 5/6], 1/r[7]], [[5/6, 1/3, 2/3, 1/3], 3/2/r[7]], [[5/6, 1/6, 1/3, 1/3], 3/2/r[7]], [[5/6, 1/6, 1/3, 4/3], 1/2/r[7]], [[5/6, 1/
6, 2/3, 1/2], 3/2/r[7]], [[5/6, 1/6, 2/3, 2/3], 4/3*r[7]], [[5/6, 1/6, 2/3, 3/2], 9/4/r[7]], [[5/6, 1/6, 2/3, 5/3], 8/5*r[7]], [[5/6, 1/6, 4/3, 1/3], 3/4/r[7]], [[5
/6, 1/6, 4/3, 4/3], 5/8/r[7]], [[5/6, 1/6, 5/3, 1/2], 3/r[7]], [[5/6, 1/6, 5/3, 2/3], 2*r[7]], [[5/6, 1/6, 5/3, 5/3], 9/5*r[7]], [[5/6, 5/6, 1/3, 2/3], 1/2/r[7]]],
[[[1/4, 1/4, 3/4, 1/4], r[8]], [[1/4, 3/4, 1/4, 1/2], -r[8]], [[1/4, 3/4, 1/4, 3/2], 4/5*r[8]], [[1/4, 3/4, 3/4, 3/4], 1/2/r[8]], [[1/4, 3/4, 5/4, 1/2], 2*r[8]], [[
1/4, 3/4, 5/4, 3/2], -(16+8*r[8])/(-5-5*r[8])], [[1/4, 5/4, 3/4, 1/4], 1/2/r[8]], [[3/4, 1/4, 3/4, 1/2], r[8]], [[3/4, 1/4, 3/4, 3/2], 8/3/r[8]], [[3/4, 1/4, 5/4, 1
/4], 1/r[8]], [[5/4, 5/4, 3/4, 1/4], 3/4/r[8]]], [[[1/5, 1/5, 3/5, 2/5], r[9]], [[1/5, 3/5, 2/5, 3/5], -r[9]/(3-r[9])], [[2/5, 1/5, 4/5, 1/5], 2/3*r[9]], [[2/5, 2/5
, 1/5, 4/5], -r[9]/(3-r[9])], [[2/5, 3/5, 4/5, 4/5], -1/2*(-3+r[9])/r[9]], [[3/5, 4/5, 1/5, 4/5], -(1-r[9])/(6-r[9])], [[4/5, 1/5, 3/5, 3/5], -r[9]/(-3+r[9])], [[4/
5, 2/5, 3/5, 2/5], -r[9]/(-3+r[9])]], [[[1/7, 1/7, 3/7, 2/7], r[10]], [[2/7, 1/7, 4/7, 1/7], 4/5*r[10]]], [[[1/7, 1/7, 3/7, 4/7], r[11]], [[1/7, 4/7, 2/7, 5/7], -r[
11]/(5-r[11])], [[2/7, 1/7, 4/7, 3/7], 4-4/5*r[11]], [[3/7, 1/7, 5/7, 2/7], 3-3/5*r[11]], [[3/7, 3/7, 2/7, 5/7], -5/(5-r[11])], [[3/7, 4/7, 2/7, 4/7], -5/(5-r[11])]
, [[4/7, 3/7, 1/7, 5/7], -5/r[11]], [[4/7, 3/7, 5/7, 3/7], -5/(-5+r[11])], [[5/7, 2/7, 1/7, 2/7], -(-10+2*r[11])/r[11]], [[5/7, 2/7, 4/7, 4/7], -5/(-5+r[11])]], [[[
1/7, 3/7, 5/7, 5/7], r[12]], [[2/7, 1/7, 5/7, 3/7], 1-1/3*r[12]]], [[[2/7, 3/7, 1/7, 5/7], r[13]], [[2/7, 4/7, 1/7, 4/7], r[13]], [[3/7, 2/7, 4/7, 4/7], -r[13]], [[
4/7, 1/7, 3/7, 5/7], -r[13]], [[4/7, 2/7, 5/7, 3/7], -1/2*r[13]], [[5/7, 1/7, 4/7, 4/7], -1/2*r[13]]], [[[2/7, 4/7, 3/7, 5/7], r[14]], [[3/7, 3/7, 4/7, 5/7], 2*r[14
]]], [[[3/7, 1/7, 5/7, 4/7], r[15]]], [[[4/7, 2/7, 1/7, 2/7], r[16]], [[5/7, 1/7, 2/7, 2/7], 3/4*r[16]]]]:
end:


#Hopefuls3C(r): A list of length 9 of equivalence classes of qudruples based on Hopefuls4(). Try:
#Hopefuls3C(r);

Hopefuls3C:=proc(r):
[[[[1/2, 1/2, 1/3, 0], r[1]], [[1/2, 1/2, 4/3, 0], 1/4*r[1]], [[2/3, 1/3, 1/2, 0], 3/4*r[1]], [[2/3, 1/3, 1/2, 1/2], 4/r[1]], [[2/3, 1/3, 3/2, 0], 1/4*r[1]]], [[[1/
2, 1/2, 2/3, 0], r[2]], [[1/2, 1/2, 5/3, 0], -1/2*r[2]], [[1/2, 1/6, 2/3, 1/3], r[2]], [[1/3, 1/6, 1/2, 1/2], 3/2*r[2]], [[1/3, 2/3, 1/2, 0], 3/2*r[2]], [[1/3, 2/3,
1/2, 1/2], -2/r[2]], [[1/3, 2/3, 3/2, 0], -1/2*r[2]], [[1/3, 2/3, 5/2, 0], -7/6*r[2]]], [[[1/3, 0, 5/3, 0], r[3]], [[2/3, 0, 4/3, 0], 3/4-1/2*r[3]], [[2/3, 2/3, 0,
0], 9/4-3/2*r[3]], [[2/3, 5/3, 0, 0], 15/2-6*r[3]], [[4/3, 1/3, 0, 0], 3/2*r[3]], [[4/3, 4/3, 0, 0], -3+15/4*r[3]], [[5/3, 5/3, 0, 0], 117/20-21/5*r[3]]], [[[1/7, 1
/7, 3/7, 2/7], r[4]], [[2/7, 1/7, 4/7, 1/7], 4/5*r[4]]], [[[1/7, 3/7, 5/7, 5/7], r[5]], [[2/7, 1/7, 5/7, 3/7], 1-1/3*r[5]]], [[[2/7, 3/7, 1/7, 5/7], r[6]], [[2/7, 4
/7, 1/7, 4/7], r[6]], [[3/7, 2/7, 4/7, 4/7], -r[6]], [[4/7, 1/7, 3/7, 5/7], -r[6]], [[4/7, 2/7, 5/7, 3/7], -1/2*r[6]], [[5/7, 1/7, 4/7, 4/7], -1/2*r[6]]], [[[2/7, 4
/7, 3/7, 5/7], r[7]], [[3/7, 3/7, 4/7, 5/7], 2*r[7]]], [[[3/7, 1/7, 5/7, 4/7], r[8]]], [[[4/7, 2/7, 1/7, 2/7], r[9]], [[5/7, 1/7, 2/7, 2/7], 3/4*r[9]]]]:
end:


#Hopefuls4C(r): A list of length 9 of equivalence classes of qudruples based on Hopefuls4(). Try:
#Hopefuls4C(r);
Hopefuls4C:=proc(r):
[[[[0, 0, 0, 0], r[1]]], [[[1/2, 0, 0, 0], r[2]], [[3/2, 0, 0, 0], 2/3-1/3*r[2]], [[5/2, 0, 0, 0], 8/5-3/5*r[2]]], [[[1/2, 1/6, 5/6, 1/6], r[3]], [[1/6, 1/6, 1/2, 1
/2], 2*r[3]], [[1/6, 2/3, 1/2, 1/2], -2/r[3]], [[1/6, 5/6, 1/2, 1/3], -2/r[3]], [[1/6, 5/6, 1/2, 5/6], 1/6/r[3]], [[1/6, 5/6, 3/2, 1/3], 6*r[3]], [[1/6, 5/6, 3/2, 4
/3], 32/7/r[3]], [[1/6, 5/6, 5/2, 1/3], 12*r[3]], [[3/2, 1/6, 5/6, 1/6], 1/2*r[3]], [[3/2, 5/6, 1/6, 5/6], 7/12/r[3]], [[5/2, 1/6, 5/6, 1/6], -(5+5*r[3])/(-8-6*r[3]
)], [[5/6, 1/3, 1/2, 1/2], 2/r[3]], [[5/6, 1/6, 1/2, 2/3], 2/r[3]], [[5/6, 5/6, 1/2, 1/2], 1/4*r[3]]], [[[1/3, 0, 2/3, 0], r[4]], [[1/3, 1/3, 0, 0], 3/2*r[4]]], [[[
1/3, 1/6, 5/6, 1/6], r[5]], [[1/3, 2/3, 1/6, 1/2], -3/r[5]], [[1/3, 2/3, 1/6, 3/2], 9/7/r[5]], [[1/3, 2/3, 1/6, 5/3], r[5]], [[1/3, 2/3, 5/6, 2/3], 1/3*r[5]], [[1/3
, 2/3, 5/6, 5/3], 7/18*r[5]], [[1/6, 1/3, 2/3, 1/3], 9/4/r[5]], [[1/6, 1/6, 2/3, 1/3], 4/3*r[5]], [[1/6, 2/3, 1/3, 2/3], -2/3*r[5]], [[1/6, 5/6, 1/3, 1/2], -2/3*r[5
]], [[1/6, 5/6, 1/3, 1/3], -3/r[5]], [[1/6, 5/6, 1/3, 3/2], r[5]], [[1/6, 5/6, 1/3, 4/3], 9/7/r[5]], [[1/6, 5/6, 2/3, 2/3], 2/9*r[5]], [[1/6, 5/6, 2/3, 5/3], 28/81*
r[5]], [[1/6, 5/6, 2/3, 5/6], 3/8/r[5]], [[1/6, 5/6, 4/3, 1/2], 8/3*r[5]], [[1/6, 5/6, 4/3, 1/3], 9/2/r[5]], [[1/6, 5/6, 4/3, 4/3], 81/28/r[5]], [[1/6, 5/6, 5/3, 2/
3], 7/9*r[5]], [[2/3, 1/3, 1/6, 4/3], 3/8/r[5]], [[2/3, 1/3, 5/6, 1/2], 4/3*r[5]], [[2/3, 1/3, 5/6, 3/2], 8/5*r[5]], [[2/3, 1/3, 5/6, 4/3], 9/4/r[5]], [[4/3, 1/6, 5
/6, 1/6], 2/3*r[5]], [[5/3, 5/6, 1/6, 5/6], 1/r[5]], [[5/6, 1/3, 2/3, 1/3], 3/2/r[5]], [[5/6, 1/6, 1/3, 1/3], 3/2/r[5]], [[5/6, 1/6, 1/3, 4/3], 1/2/r[5]], [[5/6, 1/
6, 2/3, 1/2], 3/2/r[5]], [[5/6, 1/6, 2/3, 2/3], 4/3*r[5]], [[5/6, 1/6, 2/3, 3/2], 9/4/r[5]], [[5/6, 1/6, 2/3, 5/3], 8/5*r[5]], [[5/6, 1/6, 4/3, 1/3], 3/4/r[5]], [[5
/6, 1/6, 4/3, 4/3], 5/8/r[5]], [[5/6, 1/6, 5/3, 1/2], 3/r[5]], [[5/6, 1/6, 5/3, 2/3], 2*r[5]], [[5/6, 1/6, 5/3, 5/3], 9/5*r[5]], [[5/6, 5/6, 1/3, 2/3], 1/2/r[5]]],
[[[1/4, 1/4, 3/4, 1/4], r[6]], [[1/4, 3/4, 1/4, 1/2], -r[6]], [[1/4, 3/4, 1/4, 3/2], 4/5*r[6]], [[1/4, 3/4, 3/4, 3/4], 1/2/r[6]], [[1/4, 3/4, 5/4, 1/2], 2*r[6]], [[
1/4, 3/4, 5/4, 3/2], -(16+8*r[6])/(-5-5*r[6])], [[1/4, 5/4, 3/4, 1/4], 1/2/r[6]], [[3/4, 1/4, 3/4, 1/2], r[6]], [[3/4, 1/4, 3/4, 3/2], 8/3/r[6]], [[3/4, 1/4, 5/4, 1
/4], 1/r[6]], [[5/4, 5/4, 3/4, 1/4], 3/4/r[6]]], [[[1/5, 1/5, 3/5, 2/5], r[7]], [[1/5, 3/5, 2/5, 3/5], -r[7]/(3-r[7])], [[2/5, 1/5, 4/5, 1/5], 2/3*r[7]], [[2/5, 2/5
, 1/5, 4/5], -r[7]/(3-r[7])], [[2/5, 3/5, 4/5, 4/5], -1/2*(-3+r[7])/r[7]], [[3/5, 4/5, 1/5, 4/5], -(1-r[7])/(6-r[7])], [[4/5, 1/5, 3/5, 3/5], -r[7]/(-3+r[7])], [[4/
5, 2/5, 3/5, 2/5], -r[7]/(-3+r[7])]], [[[1/7, 1/7, 3/7, 4/7], r[8]], [[1/7, 4/7, 2/7, 5/7], -r[8]/(5-r[8])], [[2/7, 1/7, 4/7, 3/7], 4-4/5*r[8]], [[3/7, 1/7, 5/7, 2/
7], 3-3/5*r[8]], [[3/7, 3/7, 2/7, 5/7], -5/(5-r[8])], [[3/7, 4/7, 2/7, 4/7], -5/(5-r[8])], [[4/7, 3/7, 1/7, 5/7], -5/r[8]], [[4/7, 3/7, 5/7, 3/7], -5/(-5+r[8])], [[
5/7, 2/7, 1/7, 2/7], -(-10+2*r[8])/r[8]], [[5/7, 2/7, 4/7, 4/7], -5/(-5+r[8])]]]:
end:



Hopefuls5C:=proc(r):
[[[[[0, 0, 1/2, 0], r[1]], [[0, 0, 1/2, -1/2], 4-2*r[1]], [[0, -1/2, 1/2, -1/2], -3+3*r[1]], [[0, 1/2, 1/2, -1/2], r[1]], [[-1/2, 0, 0, 0], 6-3*r[1]], [[-1/2, -1/2,
0, 0], -4+4*r[1]], [[-1/2, -1/2, 0, -1/2], 18-12*r[1]], [[-1/2, 1/2, 0, 0], 2*r[1]], [[-1/2, 1/2, 0, -1/2], 6-3*r[1]]]], [[[[0, 0, 1/3, -2/3], r[1]], [[0, 0, 1/3, 1
/3], 3/2+1/4*r[1]], [[0, 0, 2/3, -1/3], 3/2-1/4*r[1]], [[0, 0, 2/3, 2/3], 9/4+1/8*r[1]], [[0, -2/3, 2/3, -2/3], 15/8-5/8*r[1]], [[0, -1/3, 1/3, -1/3], r[1]], [[0, 1
/3, 2/3, -2/3], 3/2-1/4*r[1]], [[0, 2/3, 1/3, -1/3], 3/2+1/4*r[1]], [[-2/3, 0, -1/3, 0], 5/3*r[1]], [[-2/3, 0, 2/3, 0], 5/4+5/48*r[1]], [[-2/3, -2/3, 0, 0], 21/8-7/
8*r[1]], [[-2/3, -2/3, -1/3, -1/3], -35/6+35/6*r[1]], [[-2/3, -1/3, 0, -1/3], 4-2*r[1]], [[-2/3, -1/3, 0, 2/3], 5/2+1/4*r[1]], [[-2/3, 1/3, 0, 0], 3-1/2*r[1]], [[-2
/3, 1/3, -1/3, -1/3], 5/3*r[1]], [[-2/3, 2/3, 0, -1/3], 3-1/2*r[1]], [[-2/3, 2/3, 0, 2/3], -9/4-1/8*r[1]], [[-1/3, 0, 1/3, 0], 2-1/3*r[1]], [[-1/3, -2/3, 0, -2/3],
-5+5*r[1]], [[-1/3, -2/3, 0, 1/3], 1+1/2*r[1]], [[-1/3, -1/3, 0, 0], 5/4*r[1]], [[-1/3, -1/3, 1/3, -2/3], 10/3-5/3*r[1]], [[-1/3, 1/3, 0, -2/3], 5/4*r[1]], [[-1/3,
1/3, 0, 1/3], 3+1/2*r[1]], [[-1/3, 2/3, 0, 0], 3+1/2*r[1]], [[-1/3, 2/3, 1/3, -2/3], 2-1/3*r[1]], [[1/3, 0, 2/3, 0], 1+1/6*r[1]], [[1/3, -2/3, 2/3, -1/3], 2/3+1/3*r
[1]], [[1/3, 1/3, 2/3, -1/3], 1+1/6*r[1]]]], [[[[-3/4, -3/4, -1/4, -3/4], r[1]], [[-3/4, -3/4, -1/4, -1/4], 2-3/8*r[1]], [[-3/4, -3/4, -1/4, 1/4], 5/3+1/16*r[1]], [
[-3/4, -3/4, -1/4, 3/4], -(16+31*r[1])/(-32+6*r[1])], [[-3/4, -3/4, 3/4, -3/4], 10/9-1/72*r[1]], [[-3/4, -3/4, 3/4, -1/4], -(80-241*r[1])/(144+54*r[1])], [[-3/4, -3
/4, 3/4, 1/4], -(-32+34*r[1])/(-48+27*r[1])], [[-3/4, -3/4, 3/4, 3/4], -(80+75*r[1])/(-432+243*r[1])], [[-3/4, -1/2, -3/4, -1/4], 3/4+45/64*r[1]], [[-3/4, -1/2, -3/
4, 3/4], -(176+69*r[1])/(-96+6*r[1])], [[-3/4, -1/2, 1/4, -1/4], -(112+105*r[1])/(-176-5*r[1])], [[-3/4, -1/2, 1/4, 3/4], -(-80+45*r[1])/(-16+17*r[1])], [[-3/4, -1/
4, -3/4, 1/2], -(80-445*r[1])/(192-9*r[1])], [[-3/4, -1/4, -1/4, -1/4], 4/3+1/4*r[1]], [[-3/4, -1/4, -1/4, 3/4], -(-64-124*r[1])/(48+9*r[1])], [[-3/4, -1/4, 1/4, -1
/2], 3/2-3/32*r[1]], [[-3/4, -1/4, 1/4, 1/2], -(64-148*r[1])/(80-15*r[1])], [[-3/4, -1/4, 3/4, -1/4], -(48-107*r[1])/(168-63*r[1])], [[-3/4, -1/4, 3/4, 3/4], -(-544
-382*r[1])/(336+63*r[1])], [[-3/4, 1/2, -3/4, 3/4], -(80+75*r[1])/(144-81*r[1])], [[-3/4, 1/2, 1/4, -1/4], -(160-170*r[1])/(16+15*r[1])], [[-3/4, 1/2, 1/4, 3/4], -(
80+75*r[1])/(80-21*r[1])], [[-3/4, 1/4, -1/4, -3/4], 4/3+1/4*r[1]], [[-3/4, 1/4, -1/4, -1/4], -(-16-63*r[1])/(48+r[1])], [[-3/4, 1/4, -1/4, 1/4], -(80-117*r[1])/(32
-6*r[1])], [[-3/4, 1/4, 3/4, -3/4], -(32+62*r[1])/(-80-15*r[1])], [[-3/4, 1/4, 3/4, -1/4], -(-112-105*r[1])/(160+30*r[1])], [[-3/4, 1/4, 3/4, 1/4], -(112+9*r[1])/(-\
80+5*r[1])], [[-3/4, 3/4, -1/4, -1/4], -(80-117*r[1])/(32-6*r[1])], [[-3/4, 3/4, -1/4, 3/4], -(80+43*r[1])/(64+12*r[1])], [[-3/4, 3/4, 1/4, -1/2], -(160-170*r[1])/(
16+15*r[1])], [[-3/4, 3/4, 1/4, 1/2], -(16+31*r[1])/(32-6*r[1])], [[-3/4, 3/4, 3/4, -1/4], -(-176-5*r[1])/(96+18*r[1])], [[-3/4, 3/4, 3/4, 3/4], -(-160+106*r[1])/(-\
240+63*r[1])], [[-1/2, -3/4, -1/2, -1/4], -(224-462*r[1])/(272-r[1])], [[-1/2, -3/4, -1/2, 3/4], -(208-141*r[1])/(16-17*r[1])], [[-1/2, -3/4, 1/2, -1/4], -(256-96*r
[1])/(-112+7*r[1])], [[-1/2, -3/4, 1/2, 3/4], -(480+162*r[1])/(-560+147*r[1])], [[-1/2, -1/4, -1/2, 1/4], -(-1120+630*r[1])/(112+9*r[1])], [[-1/2, -1/4, 1/2, -3/4],
-(-224+126*r[1])/(48-3*r[1])], [[-1/2, -1/4, 1/2, 1/4], -(336+91*r[1])/(-320+20*r[1])], [[-1/2, 1/4, 1/2, -1/4], -(-80-75*r[1])/(112+9*r[1])], [[-1/2, 3/4, 1/2, -3/
4], -(-80-75*r[1])/(112+9*r[1])], [[-1/4, -3/4, -1/4, -1/2], 7/12+35/64*r[1]], [[-1/4, -3/4, -1/4, 1/2], 5/3+1/16*r[1]], [[-1/4, -3/4, 1/4, -3/4], 8/5-3/10*r[1]], [
[-1/4, -3/4, 1/4, 1/4], -(128-200*r[1])/(80+15*r[1])], [[-1/4, -3/4, 3/4, -1/2], 47/42-5/224*r[1]], [[-1/4, -3/4, 3/4, 1/2], -(272-197*r[1])/(168-126*r[1])], [[-1/4
, -1/2, -1/4, 1/4], -(112-119*r[1])/(32-2*r[1])], [[-1/4, -1/2, 3/4, -3/4], -(336+91*r[1])/(-416-6*r[1])], [[-1/4, -1/2, 3/4, 1/4], -(-112+63*r[1])/(-48+51*r[1])],
[[-1/4, -1/4, 1/4, -3/4], 1+3/16*r[1]], [[-1/4, -1/4, 1/4, -1/4], 3/2-3/32*r[1]], [[-1/4, -1/4, 1/4, 1/4], -(32+30*r[1])/(-48+3*r[1])], [[-1/4, -1/4, 1/4, 3/4], -(-\
48+51*r[1])/(-32+18*r[1])], [[-1/4, 1/2, 3/4, -3/4], -(16-81*r[1])/(64+12*r[1])], [[-1/4, 1/2, 3/4, 1/4], -(-64+36*r[1])/(-16+17*r[1])], [[-1/4, 1/4, -1/4, 1/2], -(
32+30*r[1])/(-16+r[1])], [[-1/4, 1/4, 1/4, -3/4], 3/2-3/32*r[1]], [[-1/4, 1/4, 1/4, 1/4], -(16-81*r[1])/(32+6*r[1])], [[-1/4, 1/4, 3/4, -1/2], -(16-81*r[1])/(64+12*
r[1])], [[-1/4, 1/4, 3/4, 1/2], -(16+31*r[1])/(-32+6*r[1])], [[-1/4, 3/4, 1/4, -3/4], -(32+30*r[1])/(-48+3*r[1])], [[-1/4, 3/4, 1/4, -1/4], -(16-81*r[1])/(32+6*r[1]
)], [[-1/4, 3/4, 1/4, 3/4], -(-64+36*r[1])/(48-51*r[1])], [[1/2, -3/4, 1/2, -1/4], -(-160+50*r[1])/(80+3*r[1])], [[1/2, -3/4, 1/2, 3/4], -(-160+10*r[1])/(48+45*r[1]
)], [[1/2, -1/4, 1/2, 1/4], -(16+15*r[1])/(-32+2*r[1])], [[1/4, -3/4, 3/4, -3/4], 1+3/80*r[1]], [[1/4, -3/4, 3/4, -1/4], -(96-150*r[1])/(80+15*r[1])], [[1/4, -3/4,
3/4, 1/4], -(-48+51*r[1])/(-80+45*r[1])], [[1/4, -3/4, 3/4, 3/4], -(80-21*r[1])/(-160+90*r[1])], [[1/4, -1/2, 1/4, -1/4], -(80-85*r[1])/(32-2*r[1])], [[1/4, -1/2, 1
/4, 3/4], -(80+75*r[1])/(-160+42*r[1])], [[1/4, -1/4, 1/4, 1/2], -(32+30*r[1])/(-48+3*r[1])], [[1/4, -1/4, 3/4, -1/4], -(-48+19*r[1])/(16+3*r[1])], [[1/4, -1/4, 3/4
, 3/4], -(80+43*r[1])/(-64-12*r[1])], [[1/4, 1/2, 1/4, 3/4], -(-48+51*r[1])/(64-36*r[1])], [[1/4, 1/4, 3/4, -3/4], -(-48+19*r[1])/(16+3*r[1])], [[1/4, 1/4, 3/4, -1/
4], -(16-81*r[1])/(64+12*r[1])], [[1/4, 1/4, 3/4, 1/4], -(48-51*r[1])/(64-36*r[1])], [[1/4, 3/4, 3/4, -1/4], -(48-51*r[1])/(64-36*r[1])], [[1/4, 3/4, 3/4, 3/4], -(-\
32+18*r[1])/(-48+51*r[1])], [[3/4, -3/4, 3/4, -1/2], -(-112+55*r[1])/(16+15*r[1])], [[3/4, -3/4, 3/4, 1/2], -(16+15*r[1])/(-144+81*r[1])], [[3/4, -1/2, 3/4, 1/4], -
(32-18*r[1])/(16-17*r[1])], [[3/4, 1/4, 3/4, 1/2], -(48-51*r[1])/(64-36*r[1])]]], [[[[-4/5, -4/5, -2/5, -3/5], r[1]], [[-4/5, -4/5, -2/5, 2/5], 13/7+3/70*r[1]], [[-\
4/5, -4/5, -1/5, -1/5], 13/7-9/35*r[1]], [[-4/5, -4/5, -1/5, 4/5], -(195+322*r[1])/(-305+32*r[1])], [[-4/5, -4/5, 3/5, -3/5], 13/11-1/55*r[1]], [[-4/5, -4/5, 3/5, 2
/5], -(975+1020*r[1])/(-2035+264*r[1])], [[-4/5, -4/5, 4/5, -1/5], -(-455+303*r[1])/(90-48*r[1])], [[-4/5, -4/5, 4/5, 4/5], -(845-348*r[1])/(80-192*r[1])], [[-4/5,
-3/5, -1/5, -2/5], 2-24/65*r[1]], [[-4/5, -3/5, -1/5, 3/5], -(650-480*r[1])/(65-36*r[1])], [[-4/5, -3/5, 4/5, -2/5], -(-130-612*r[1])/(715+132*r[1])], [[-4/5, -3/5,
4/5, 3/5], -(-520+528*r[1])/(-715+396*r[1])], [[-4/5, -2/5, -3/5, -2/5], 55/42+22/91*r[1]], [[-4/5, -2/5, -3/5, 3/5], -(-715+176*r[1])/(130+24*r[1])], [[-4/5, -2/5,
2/5, -2/5], -(-2145-627*r[1])/(2210+72*r[1])], [[-4/5, -2/5, 2/5, 3/5], -(715+594*r[1])/(-1690+696*r[1])], [[-4/5, -1/5, -3/5, -3/5], 55/42+22/91*r[1]], [[-4/5, -1/
5, -3/5, 2/5], -(325+312*r[1])/(-299+12*r[1])], [[-4/5, -1/5, -2/5, -1/5], 10/7+24/91*r[1]], [[-4/5, -1/5, -2/5, 4/5], -(650-678*r[1])/(13+36*r[1])], [[-4/5, -1/5,
2/5, -3/5], -(65-204*r[1])/(156+12*r[1])], [[-4/5, -1/5, 2/5, 2/5], -(585+82*r[1])/(-377+20*r[1])], [[-4/5, -1/5, 3/5, -1/5], -(65+138*r[1])/(-234+24*r[1])], [[-4/5
, -1/5, 3/5, 4/5], -(-1040-381*r[1])/(377+36*r[1])], [[-4/5, 1/5, -2/5, -3/5], 10/7+24/91*r[1]], [[-4/5, 1/5, -2/5, 2/5], -(585+836*r[1])/(-455+28*r[1])], [[-4/5, 1
/5, -1/5, -1/5], 11/5-36/325*r[1]], [[-4/5, 1/5, 3/5, -3/5], -(715-316*r[1])/(-260+36*r[1])], [[-4/5, 1/5, 3/5, 2/5], -(-845+348*r[1])/(65+54*r[1])], [[-4/5, 1/5, 4
/5, -1/5], -(260-464*r[1])/(195+92*r[1])], [[-4/5, 2/5, -1/5, -2/5], 11/5-36/325*r[1]], [[-4/5, 2/5, 4/5, -2/5], -(-715+456*r[1])/(260-144*r[1])], [[-4/5, 3/5, -3/5
, -2/5], -(325+312*r[1])/(-299+12*r[1])], [[-4/5, 3/5, -3/5, 3/5], -(65+264*r[1])/(-325+276*r[1])], [[-4/5, 3/5, 2/5, -2/5], -(65+54*r[1])/(-65-12*r[1])], [[-4/5, 3
/5, 2/5, 3/5], -(325+60*r[1])/(195-48*r[1])], [[-4/5, 4/5, -3/5, -3/5], -(-715+176*r[1])/(130+24*r[1])], [[-4/5, 4/5, -3/5, 2/5], -(65+264*r[1])/(-325+276*r[1])], [
[-4/5, 4/5, -2/5, -1/5], -(585+836*r[1])/(-455+28*r[1])], [[-4/5, 4/5, -2/5, 4/5], -(585+276*r[1])/(520+96*r[1])], [[-4/5, 4/5, 2/5, -3/5], -(65+54*r[1])/(-65-12*r[
1])], [[-4/5, 4/5, 2/5, 2/5], -(195-293*r[1])/(-325+164*r[1])], [[-4/5, 4/5, 3/5, -1/5], -(130-102*r[1])/(195-132*r[1])], [[-3/5, -4/5, -1/5, -4/5], 12/13*r[1]], [[
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-2/5], -(-1430+660*r[1])/(325+4*r[1])], [[-3/5, -3/5, -2/5, 3/5], -(2145+1012*r[1])/(-1820+168*r[1])], [[-3/5, -3/5, 1/5, -1/5], -(1430-462*r[1])/(-585+18*r[1])], [
[-3/5, -3/5, 1/5, 4/5], -(715-396*r[1])/(520-408*r[1])], [[-3/5, -3/5, 3/5, -2/5], 616*r[1]/(585+108*r[1])], [[-3/5, -3/5, 3/5, 3/5], -(-6435+44*r[1])/(2925+540*r[1
])], [[-3/5, -2/5, -1/5, -1/5], 5/4+3/13*r[1]], [[-3/5, -2/5, -1/5, 4/5], -(325-752*r[1])/(143+60*r[1])], [[-3/5, -2/5, 1/5, -2/5], 10/7-4/91*r[1]], [[-3/5, -2/5, 1
/5, 3/5], -(-845+852*r[1])/(-468+216*r[1])], [[-3/5, -2/5, 4/5, -1/5], -(-520+212*r[1])/(39+108*r[1])], [[-3/5, -2/5, 4/5, 4/5], -(390-572*r[1])/(39+108*r[1])], [[-\
3/5, -1/5, -2/5, -4/5], -(-1430+660*r[1])/(325+4*r[1])], [[-3/5, -1/5, -2/5, 1/5], -(-780+612*r[1])/(-65+16*r[1])], [[-3/5, -1/5, 3/5, -4/5], -(780-72*r[1])/(-585+4
*r[1])], [[-3/5, -1/5, 3/5, 1/5], -(-715+516*r[1])/(-325+220*r[1])], [[-3/5, 1/5, -1/5, -4/5], 5/4+3/13*r[1]], [[-3/5, 1/5, -1/5, 1/5], -(-195+608*r[1])/(-260+12*r[
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65+16*r[1])], [[-3/5, 2/5, 1/5, -1/5], -(65-156*r[1])/(65+12*r[1])], [[-3/5, 2/5, 1/5, 4/5], -(325+60*r[1])/(130-32*r[1])], [[-3/5, 2/5, 3/5, -2/5], -(-260-216*r[1]
)/(325+60*r[1])], [[-3/5, 3/5, -1/5, -1/5], -(-195+608*r[1])/(-260+12*r[1])], [[-3/5, 3/5, -1/5, 4/5], -(260+216*r[1])/(195+36*r[1])], [[-3/5, 3/5, 1/5, -2/5], -(65
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, -(2145+1012*r[1])/(-1820+168*r[1])], [[-3/5, 4/5, 3/5, -4/5], -(-260-216*r[1])/(325+60*r[1])], [[-2/5, -4/5, 2/5, -1/5], -(715-792*r[1])/(260+48*r[1])], [[-2/5, -\
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5], -(-325+10*r[1])/(195+36*r[1])], [[-2/5, -2/5, -1/5, -4/5], 15/14+18/91*r[1]], [[-2/5, -2/5, -1/5, 1/5], -(65+264*r[1])/(-260+8*r[1])], [[-2/5, -2/5, 2/5, -3/5],
-(130-270*r[1])/(195+8*r[1])], [[-2/5, -2/5, 2/5, 2/5], -(65+54*r[1])/(-130+32*r[1])], [[-2/5, -2/5, 4/5, -4/5], -(780-297*r[1])/(-390+40*r[1])], [[-2/5, -2/5, 4/5,
1/5], -(-715+246*r[1])/(-455+532*r[1])], [[-2/5, -1/5, 1/5, -1/5], -(-130-248*r[1])/(325+4*r[1])], [[-2/5, -1/5, 1/5, 4/5], -(-65+156*r[1])/(-65+16*r[1])], [[-2/5,
1/5, 2/5, -1/5], -(325+60*r[1])/(-260-6*r[1])], [[-2/5, 2/5, -1/5, -3/5], -(65+264*r[1])/(-260+8*r[1])], [[-2/5, 2/5, -1/5, 2/5], -(-780+192*r[1])/(65+12*r[1])], [[
-2/5, 2/5, 1/5, -4/5], -(-130-248*r[1])/(325+4*r[1])], [[-2/5, 2/5, 1/5, 1/5], -(-845-72*r[1])/(325+4*r[1])], [[-2/5, 2/5, 4/5, -3/5], -(-130-108*r[1])/(195+36*r[1]
)], [[-2/5, 2/5, 4/5, 2/5], -(195-293*r[1])/(325-164*r[1])], [[-2/5, 3/5, -1/5, -4/5], -(650-608*r[1])/(169-36*r[1])], [[-2/5, 3/5, -1/5, 1/5], -(-780+192*r[1])/(65
+12*r[1])], [[-2/5, 3/5, 2/5, -3/5], -(325+60*r[1])/(-260-6*r[1])], [[-2/5, 3/5, 4/5, -4/5], -(-130-108*r[1])/(195+36*r[1])], [[-2/5, 3/5, 4/5, 1/5], -(65+124*r[1])
/(-325+150*r[1])], [[-2/5, 4/5, 1/5, -1/5], -(-845-72*r[1])/(325+4*r[1])], [[-2/5, 4/5, 1/5, 4/5], -(390-96*r[1])/(325+60*r[1])], [[-1/5, -4/5, 2/5, -4/5], 10/7-18/
91*r[1]], [[-1/5, -4/5, 2/5, 1/5], -(-130-156*r[1])/(247+12*r[1])], [[-1/5, -4/5, 3/5, -2/5], -(260+174*r[1])/(-429-12*r[1])], [[-1/5, -4/5, 3/5, 3/5], -(130+31*r[1
])/(-169+36*r[1])], [[-1/5, -3/5, 3/5, -3/5], -(130-42*r[1])/(-65+2*r[1])], [[-1/5, -3/5, 3/5, 2/5], -(-195+108*r[1])/(-260+204*r[1])], [[-1/5, -2/5, 1/5, -3/5], 1+
12/65*r[1]], [[-1/5, -2/5, 1/5, 2/5], -(-195+188*r[1])/(-65+16*r[1])], [[-1/5, -1/5, 1/5, -4/5], 1+12/65*r[1]], [[-1/5, -1/5, 1/5, 1/5], -(-325+360*r[1])/(-130+12*r
[1])], [[-1/5, -1/5, 2/5, -2/5], -(65-226*r[1])/(195+4*r[1])], [[-1/5, -1/5, 2/5, 3/5], -(260-204*r[1])/(65-36*r[1])], [[-1/5, 1/5, 2/5, -4/5], -(65-226*r[1])/(195+
4*r[1])], [[-1/5, 1/5, 2/5, 1/5], -(65-156*r[1])/(65+12*r[1])], [[-1/5, 1/5, 3/5, -2/5], -(715-316*r[1])/(-260+36*r[1])], [[-1/5, 1/5, 3/5, 3/5], -(585+136*r[1])/(-\
260+36*r[1])], [[-1/5, 2/5, 3/5, -3/5], -(715-316*r[1])/(-260+36*r[1])], [[-1/5, 2/5, 3/5, 2/5], -(325+60*r[1])/(-195+48*r[1])], [[-1/5, 3/5, 1/5, -3/5], -(-325+360
*r[1])/(-130+12*r[1])], [[-1/5, 4/5, 1/5, -4/5], -(-195+188*r[1])/(-65+16*r[1])], [[-1/5, 4/5, 2/5, -2/5], -(65-156*r[1])/(65+12*r[1])], [[-1/5, 4/5, 2/5, 3/5], -(
390-96*r[1])/(325+60*r[1])], [[1/5, -4/5, 3/5, -3/5], 1+4/65*r[1]], [[1/5, -4/5, 3/5, 2/5], -(-65+156*r[1])/(-195+48*r[1])], [[1/5, -4/5, 4/5, -1/5], -(-195+44*r[1]
)/(130-4*r[1])], [[1/5, -4/5, 4/5, 4/5], -(-195+108*r[1])/(-260+204*r[1])], [[1/5, -3/5, 4/5, -2/5], -(455-504*r[1])/(195+36*r[1])], [[1/5, -3/5, 4/5, 3/5], -(-65+
156*r[1])/(-585+324*r[1])], [[1/5, -2/5, 2/5, -2/5], -(65+264*r[1])/(-390+12*r[1])], [[1/5, -2/5, 2/5, 3/5], -(325+60*r[1])/(-390+96*r[1])], [[1/5, -1/5, 2/5, -3/5]
, -(65+264*r[1])/(-390+12*r[1])], [[1/5, -1/5, 2/5, 2/5], -(-195+48*r[1])/(65+12*r[1])], [[1/5, -1/5, 3/5, -1/5], -(65+264*r[1])/(-390+12*r[1])], [[1/5, -1/5, 3/5,
4/5], -(260+216*r[1])/(-195-36*r[1])], [[1/5, 1/5, 3/5, -3/5], -(65+264*r[1])/(-390+12*r[1])], [[1/5, 1/5, 3/5, 2/5], -(-65+156*r[1])/(-130+32*r[1])], [[1/5, 1/5, 4
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/5], -(-130+102*r[1])/(-65+36*r[1])], [[2/5, -3/5, 3/5, -2/5], -(-260+120*r[1])/(65+12*r[1])], [[2/5, -3/5, 3/5, 3/5], -(-845-72*r[1])/(195+372*r[1])], [[2/5, -2/5,
4/5, -1/5], -(-390+26*r[1])/(325+4*r[1])], [[2/5, -2/5, 4/5, 4/5], -(585+276*r[1])/(-520-96*r[1])], [[2/5, -1/5, 3/5, -4/5], -(-260+120*r[1])/(65+12*r[1])], [[2/5,
-1/5, 3/5, 1/5], -(-65+156*r[1])/(-195+48*r[1])], [[2/5, 1/5, 4/5, -4/5], -(-390+26*r[1])/(325+4*r[1])], [[2/5, 1/5, 4/5, 1/5], -(-65+156*r[1])/(-195+48*r[1])], [[2
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1])], [[3/5, -2/5, 4/5, -4/5], -(-130+32*r[1])/(65+12*r[1])], [[3/5, -2/5, 4/5, 1/5], -(260-64*r[1])/(65-156*r[1])], [[3/5, 2/5, 4/5, -3/5], -(260-64*r[1])/(65-156*
r[1])], [[3/5, 2/5, 4/5, 2/5], -(325+60*r[1])/(-390+96*r[1])], [[3/5, 3/5, 4/5, -4/5], -(260+6*r[1])/(-195-36*r[1])], [[3/5, 3/5, 4/5, 1/5], -(325+60*r[1])/(-390+96
*r[1])]]], [[[[0, -1/2, 1/6, -1/2], r[1]], [[0, -1/2, 1/6, 1/2], 3/2+1/8*r[1]], [[0, -1/3, 1/6, -2/3], r[1]], [[0, -1/3, 1/6, 1/3], 4/3+2/9*r[1]], [[0, 1/2, 1/6, -1
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/45*r[1]], [[-5/6, 1/2, 0, -1/2], 12/5-2/5*r[1]], [[-5/6, 1/3, 0, -1/3], 12/5-2/5*r[1]], [[-2/3, -5/6, -1/2, -1/2], -6+6*r[1]], [[-2/3, -5/6, -1/2, 1/2], 3/2+3/8*r[
1]], [[-2/3, -5/6, 1/2, -1/2], 3/2-1/4*r[1]], [[-2/3, -5/6, 1/2, 1/2], 3/2+1/16*r[1]], [[-2/3, -1/3, -1/2, 0], 3/2*r[1]], [[-2/3, -1/3, -1/2, 1/2], -(-72+6*r[1])/(
24+r[1])], [[-2/3, -1/3, 1/2, -1/2], -(24+6*r[1])/(-24-r[1])], [[-2/3, -1/3, 1/2, 1/2], 48/(24+r[1])], [[-2/3, 1/6, -1/2, -1/2], 3/2*r[1]], [[-2/3, 1/6, -1/2, 1/2],
9/2+3/8*r[1]], [[-2/3, 1/6, 1/2, -1/2], 3/2-1/8*r[1]], [[-2/3, 1/6, 1/2, 1/2], 3+1/8*r[1]], [[-2/3, 2/3, -1/2, 0], 9/2+3/8*r[1]], [[-2/3, 2/3, -1/2, -1/2], -(-72+6*
r[1])/(24+r[1])], [[-2/3, 2/3, 1/2, -1/2], -(-36-3*r[1])/(24+r[1])], [[-2/3, 2/3, 1/2, 1/2], -72/(24+r[1])], [[-1/2, 0, 1/3, -1/3], 9/5-3/10*r[1]], [[-1/2, 0, 1/3,
2/3], 18/5+3/20*r[1]], [[-1/2, -5/6, -1/3, -2/3], -28/5+28/5*r[1]], [[-1/2, -5/6, -1/3, 1/3], 14/15+28/45*r[1]], [[-1/2, -5/6, 2/3, -2/3], 7/5-7/30*r[1]], [[-1/2, -\
5/6, 2/3, 1/3], 56/45+7/135*r[1]], [[-1/2, -1/2, -1/3, 0], 4/3*r[1]], [[-1/2, -1/2, -1/3, 1/2], -(-60+5*r[1])/(24+r[1])], [[-1/2, -1/2, 1/3, -5/6], 27/10-6/5*r[1]],
[[-1/2, -1/2, 1/3, -1/3], -(18+12*r[1])/(-24-r[1])], [[-1/2, -1/2, 1/3, 2/3], 54/(24+r[1])], [[-1/2, -1/2, 2/3, 0], 6/5+1/30*r[1]], [[-1/2, -1/2, 2/3, -1/2], -(-24-\
4*r[1])/(24+r[1])], [[-1/2, -1/2, 2/3, 1/2], -(-192-2*r[1])/(120+5*r[1])], [[-1/2, -1/3, -1/3, 1/3], -(-66+11*r[1])/(24+r[1])], [[-1/2, -1/3, 2/3, -2/3], -(-44-11*r
[1])/(48+2*r[1])], [[-1/2, -1/3, 2/3, 1/3], 110/(72+3*r[1])], [[-1/2, 1/2, -1/3, 0], 8/3+4/9*r[1]], [[-1/2, 1/2, -1/3, -1/2], -(-66+11*r[1])/(24+r[1])], [[-1/2, 1/2
, 1/3, -5/6], 9/5-3/10*r[1]], [[-1/2, 1/2, 1/3, -1/3], -(-36-6*r[1])/(24+r[1])], [[-1/2, 1/2, 1/3, 2/3], -108/(24+r[1])], [[-1/2, 1/2, 2/3, -1/2], -(60+5*r[1])/(-48
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1]], [[-1/2, 2/3, -1/3, -2/3], -(-60+5*r[1])/(24+r[1])], [[-1/2, 2/3, 2/3, -2/3], -(60+5*r[1])/(-48-2*r[1])], [[-1/3, 0, 1/2, -1/2], 8/5-4/15*r[1]], [[-1/3, 0, 1/2,
1/2], 32/15+4/45*r[1]], [[-1/3, -2/3, 1/2, -5/6], 32/15-4/5*r[1]], [[-1/3, -1/2, 1/2, -1/2], -(33+22*r[1])/(-48-2*r[1])], [[-1/3, -1/2, 1/2, 1/2], 165/(96+4*r[1])],
[[-1/3, 1/2, 1/2, -1/2], -(-30-5*r[1])/(24+r[1])], [[-1/3, 1/3, 1/2, -5/6], 8/5-4/15*r[1]], [[-1/3, 1/3, 1/2, -1/3], -(-30-5*r[1])/(24+r[1])], [[1/2, -5/6, 2/3, -2/
3], 4/5+1/5*r[1]], [[1/2, -5/6, 2/3, 1/3], 16/15+2/45*r[1]], [[1/2, -1/2, 2/3, 0], 1+1/12*r[1]], [[1/2, -1/2, 2/3, -1/2], -(-60+5*r[1])/(48+2*r[1])], [[1/2, -1/3, 2
/3, -2/3], -(-60+5*r[1])/(48+2*r[1])], [[1/2, -1/3, 2/3, 1/3], 30/(24+r[1])], [[1/2, 1/2, 2/3, 0], 4/3+1/18*r[1]], [[1/2, 1/2, 2/3, -1/2], 30/(24+r[1])], [[1/2, 1/6
, 2/3, -2/3], 1+1/12*r[1]], [[1/2, 1/6, 2/3, 1/3], 4/3+1/18*r[1]], [[1/3, -5/6, 1/2, -1/2], 3/5+2/5*r[1]], [[1/3, -5/6, 1/2, 1/2], 6/5+1/20*r[1]], [[1/3, -1/3, 1/2,
0], 1+1/6*r[1]], [[1/3, -1/3, 1/2, -1/2], -(-36+6*r[1])/(24+r[1])], [[1/3, -1/3, 1/2, 1/2], 36/(24+r[1])], [[1/3, 1/6, 1/2, -1/2], 1+1/6*r[1]], [[1/3, 1/6, 1/2, 1/2
], 2+1/12*r[1]], [[1/3, 2/3, 1/2, 0], 2+1/12*r[1]], [[1/3, 2/3, 1/2, -1/2], 36/(24+r[1])], [[1/3, 2/3, 1/2, 1/2], -36/(24+r[1])]], [[[-5/6, -5/6, -1/2, -1/2], r[2]]
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, [[-3/7, -2/7, -1/7, -1/7], 18/5-108/85*r[5]], [[-3/7, -2/7, -1/7, 6/7], 58/15+4/85*r[5]], [[-3/7, -2/7, 6/7, -1/7], 62/55-12/935*r[5]], [[-3/7, -2/7, 6/7, 6/7], 
412/165+16/935*r[5]], [[-3/7, 1/7, -1/7, -4/7], 18/5-108/85*r[5]], [[-3/7, 1/7, -1/7, 3/7], 19/5-24/85*r[5]], [[-3/7, 1/7, 6/7, -4/7], 21/20+9/340*r[5]], [[-3/7, 1/
7, 6/7, 3/7], 8/5-3/85*r[5]], [[-3/7, 5/7, -1/7, -1/7], 19/5-24/85*r[5]], [[-3/7, 5/7, -1/7, 6/7], -103/90-2/255*r[5]], [[-3/7, 5/7, 6/7, -1/7], 61/45-2/255*r[5]],
[[-3/7, 5/7, 6/7, 6/7], 19/180-2/255*r[5]], [[-2/7, -3/7, 3/7, -6/7], -1/2+18/17*r[5]], [[-2/7, -3/7, 3/7, 1/7], 13/10+6/85*r[5]], [[-2/7, -1/7, 3/7, -1/7], 1+4/17*
r[5]], [[-2/7, -1/7, 3/7, 6/7], 23/5-8/85*r[5]], [[-2/7, 4/7, 3/7, -6/7], 1+4/17*r[5]], [[-2/7, 4/7, 3/7, 1/7], 14/5+6/85*r[5]], [[-2/7, 6/7, 3/7, -1/7], 14/5+6/85*
r[5]], [[-2/7, 6/7, 3/7, 6/7], -19/125+24/2125*r[5]], [[-1/7, -4/7, 4/7, -6/7], 12/17*r[5]], [[-1/7, -4/7, 4/7, 1/7], 6/5+4/85*r[5]], [[-1/7, -2/7, 1/7, -3/7], 3-18
/17*r[5]], [[-1/7, -2/7, 1/7, 4/7], 11/5-6/85*r[5]], [[-1/7, -1/7, 1/7, -4/7], 3-18/17*r[5]], [[-1/7, -1/7, 1/7, 3/7], 57/25-72/425*r[5]], [[-1/7, -1/7, 4/7, -2/7],
9/10+18/85*r[5]], [[-1/7, -1/7, 4/7, 5/7], 66/25-36/425*r[5]], [[-1/7, 3/7, 4/7, -6/7], 9/10+18/85*r[5]], [[-1/7, 3/7, 4/7, 1/7], 28/15+4/85*r[5]], [[-1/7, 5/7, 1/7
, -3/7], 57/25-72/425*r[5]], [[-1/7, 5/7, 1/7, 4/7], -47/25+12/425*r[5]], [[-1/7, 6/7, 1/7, -4/7], 11/5-6/85*r[5]], [[-1/7, 6/7, 1/7, 3/7], -47/25+12/425*r[5]], [[-\
1/7, 6/7, 4/7, -2/7], 28/15+4/85*r[5]], [[-1/7, 6/7, 4/7, 5/7], 3/50+6/425*r[5]], [[1/7, -4/7, 6/7, -1/7], 1+2/51*r[5]], [[1/7, -4/7, 6/7, 6/7], 8/5-4/255*r[5]], [[
1/7, -3/7, 6/7, -2/7], 39/40+9/170*r[5]], [[1/7, -3/7, 6/7, 5/7], 141/100-9/425*r[5]], [[1/7, 3/7, 6/7, -1/7], 28/25+12/425*r[5]], [[1/7, 3/7, 6/7, 6/7], -19/50+12/
425*r[5]], [[1/7, 4/7, 6/7, -2/7], 28/25+12/425*r[5]], [[1/7, 4/7, 6/7, 5/7], 3/25+12/425*r[5]], [[3/7, -6/7, 5/7, -3/7], 4/3-3/17*r[5]], [[3/7, -6/7, 5/7, 4/7], 6/
5-1/85*r[5]], [[3/7, -1/7, 5/7, -1/7], 19/15-8/85*r[5]], [[3/7, -1/7, 5/7, 6/7], 103/45+4/255*r[5]], [[3/7, 1/7, 5/7, -3/7], 19/15-8/85*r[5]], [[3/7, 1/7, 5/7, 4/7]
, 47/25-12/425*r[5]], [[3/7, 6/7, 5/7, -1/7], 47/25-12/425*r[5]], [[3/7, 6/7, 5/7, 6/7], 197/375-4/2125*r[5]], [[4/7, -6/7, 6/7, -4/7], 9/8-9/136*r[5]], [[4/7, -6/7
, 6/7, 3/7], 27/25-9/850*r[5]], [[4/7, -2/7, 6/7, -1/7], 11/10-3/85*r[5]], [[4/7, -2/7, 6/7, 6/7], 89/60+1/170*r[5]], [[4/7, 1/7, 6/7, -4/7], 11/10-3/85*r[5]], [[4/
7, 1/7, 6/7, 3/7], 94/75-8/425*r[5]], [[4/7, 5/7, 6/7, -1/7], 94/75-8/425*r[5]], [[4/7, 5/7, 6/7, 6/7], 197/300-1/425*r[5]]], [[[-6/7, -3/7, -5/7, -3/7], r[6]], [[-\
6/7, -3/7, -5/7, 4/7], 8/3+1/15*r[6]], [[-6/7, -3/7, 2/7, -3/7], 16/11-2/33*r[6]], [[-6/7, -3/7, 2/7, 4/7], 73/33+5/264*r[6]], [[-6/7, -2/7, -5/7, -4/7], r[6]], [[-\
6/7, -2/7, -5/7, 3/7], 5/2+1/8*r[6]], [[-6/7, -2/7, 2/7, -4/7], 3/2-3/32*r[6]], [[-6/7, -2/7, 2/7, 3/7], 37/18+1/36*r[6]], [[-6/7, 4/7, -5/7, -3/7], 5/2+1/8*r[6]],
[[-6/7, 4/7, -5/7, 4/7], -10-1/20*r[6]], [[-6/7, 4/7, 2/7, -3/7], 11/6-1/48*r[6]], [[-6/7, 4/7, 2/7, 4/7], -89/16-5/128*r[6]], [[-6/7, 5/7, -5/7, -4/7], 8/3+1/15*r[
6]], [[-6/7, 5/7, -5/7, 3/7], -10-1/20*r[6]], [[-6/7, 5/7, 2/7, -4/7], 11/6-1/48*r[6]], [[-6/7, 5/7, 2/7, 3/7], -128/27-1/27*r[6]], [[-5/7, -4/7, 1/7, -2/7], 8/5-3/
25*r[6]], [[-5/7, -4/7, 1/7, 5/7], 12/5+1/50*r[6]], [[-5/7, -3/7, 1/7, -3/7], 5/3-1/6*r[6]], [[-5/7, -3/7, 1/7, 4/7], 20/9+1/36*r[6]], [[-5/7, 3/7, 1/7, -2/7], 2-1/
20*r[6]], [[-5/7, 3/7, 1/7, 5/7], -7-1/20*r[6]], [[-5/7, 4/7, 1/7, -3/7], 2-1/20*r[6]], [[-5/7, 4/7, 1/7, 4/7], -6-1/20*r[6]], [[-4/7, -5/7, -3/7, -3/7], 13/15*r[6]
], [[-4/7, -5/7, -3/7, 4/7], 2+1/20*r[6]], [[-4/7, -5/7, 4/7, -3/7], 16/13-2/65*r[6]], [[-4/7, -5/7, 4/7, 4/7], 21/13+1/104*r[6]], [[-4/7, -2/7, -3/7, -6/7], 13/15*
r[6]], [[-4/7, -2/7, -3/7, 1/7], 13/8+13/64*r[6]], [[-4/7, -2/7, 4/7, -6/7], 13/10-13/160*r[6]], [[-4/7, -2/7, 4/7, 1/7], 169/120+13/960*r[6]], [[-4/7, 2/7, -3/7, -\
3/7], 13/8+13/64*r[6]], [[-4/7, 2/7, -3/7, 4/7], 39/4+3/32*r[6]], [[-4/7, 2/7, 4/7, -3/7], 11/8-1/64*r[6]], [[-4/7, 2/7, 4/7, 4/7], 89/16+5/128*r[6]], [[-4/7, 5/7,
-3/7, -6/7], 2+1/20*r[6]], [[-4/7, 5/7, -3/7, 1/7], 39/4+3/32*r[6]], [[-4/7, 5/7, 4/7, -6/7], 11/8-1/64*r[6]], [[-4/7, 5/7, 4/7, 1/7], 25/6+1/48*r[6]], [[-3/7, -6/7
, 3/7, -2/7], 4/3-1/15*r[6]], [[-3/7, -6/7, 3/7, 5/7], 16/9+1/90*r[6]], [[-3/7, -5/7, -2/7, -4/7], 13/16*r[6]], [[-3/7, -5/7, -2/7, 3/7], 5/3+1/12*r[6]], [[-3/7, -5
/7, 5/7, -4/7], 15/13-3/104*r[6]], [[-3/7, -5/7, 5/7, 3/7], 155/117+1/117*r[6]], [[-3/7, -3/7, -2/7, -6/7], 13/16*r[6]], [[-3/7, -3/7, -2/7, 1/7], 13/9+13/72*r[6]],
[[-3/7, -3/7, 3/7, -5/7], 13/9-13/90*r[6]], [[-3/7, -3/7, 3/7, 2/7], 247/162+13/1296*r[6]], [[-3/7, -3/7, 5/7, -6/7], 13/11-13/264*r[6]], [[-3/7, -3/7, 5/7, 1/7], 
247/198+13/1584*r[6]], [[-3/7, 1/7, 3/7, -2/7], 3/2-3/80*r[6]], [[-3/7, 1/7, 3/7, 5/7], 7+1/20*r[6]], [[-3/7, 2/7, -2/7, -4/7], 13/9+13/72*r[6]], [[-3/7, 2/7, -2/7,
3/7], 14/3+1/12*r[6]], [[-3/7, 2/7, 5/7, -4/7], 11/9-1/72*r[6]], [[-3/7, 2/7, 5/7, 3/7], 64/27+1/54*r[6]], [[-3/7, 4/7, -2/7, -6/7], 5/3+1/12*r[6]], [[-3/7, 4/7, -2
/7, 1/7], 14/3+1/12*r[6]], [[-3/7, 4/7, 3/7, -5/7], 3/2-3/80*r[6]], [[-3/7, 4/7, 3/7, 2/7], 31/6+1/48*r[6]], [[-3/7, 4/7, 5/7, -6/7], 11/9-1/72*r[6]], [[-3/7, 4/7,
5/7, 1/7], 25/12+1/96*r[6]], [[-2/7, -6/7, 4/7, -3/7], 5/4-1/16*r[6]], [[-2/7, -6/7, 4/7, 4/7], 35/24+1/96*r[6]], [[-2/7, -4/7, 4/7, -5/7], 13/10-39/400*r[6]], [[-2
/7, -4/7, 4/7, 2/7], 65/48+13/1920*r[6]], [[-2/7, 1/7, 4/7, -3/7], 4/3-1/30*r[6]], [[-2/7, 1/7, 4/7, 4/7], 3+1/40*r[6]], [[-2/7, 3/7, 4/7, -5/7], 4/3-1/30*r[6]], [[
-2/7, 3/7, 4/7, 2/7], 31/12+1/96*r[6]], [[1/7, -3/7, 2/7, -3/7], 1+1/8*r[6]], [[1/7, -3/7, 2/7, 4/7], 13/8+1/64*r[6]], [[1/7, -2/7, 2/7, -4/7], 1+1/8*r[6]], [[1/7,
-2/7, 2/7, 3/7], 14/9+1/36*r[6]], [[1/7, 4/7, 2/7, -3/7], 14/9+1/36*r[6]], [[1/7, 4/7, 2/7, 4/7], -19/6-1/48*r[6]], [[1/7, 5/7, 2/7, -4/7], 13/8+1/64*r[6]], [[1/7,
5/7, 2/7, 3/7], -19/6-1/48*r[6]], [[3/7, -5/7, 4/7, -3/7], 1+1/20*r[6]], [[3/7, -5/7, 4/7, 4/7], 5/4+1/160*r[6]], [[3/7, -2/7, 4/7, -6/7], 1+1/20*r[6]], [[3/7, -2/7
, 4/7, 1/7], 7/6+1/48*r[6]], [[3/7, 2/7, 4/7, -3/7], 7/6+1/48*r[6]], [[3/7, 2/7, 4/7, 4/7], 19/6+1/48*r[6]], [[3/7, 5/7, 4/7, -6/7], 5/4+1/160*r[6]], [[3/7, 5/7, 4/
7, 1/7], 19/6+1/48*r[6]], [[4/7, -5/7, 5/7, -4/7], 1+1/40*r[6]], [[4/7, -5/7, 5/7, 3/7], 10/9+1/180*r[6]], [[4/7, -3/7, 5/7, -6/7], 1+1/40*r[6]], [[4/7, -3/7, 5/7,
1/7], 13/12+1/96*r[6]], [[4/7, 2/7, 5/7, -4/7], 13/12+1/96*r[6]], [[4/7, 2/7, 5/7, 3/7], 19/12+1/96*r[6]], [[4/7, 4/7, 5/7, -6/7], 10/9+1/180*r[6]], [[4/7, 4/7, 5/7
, 1/7], 19/12+1/96*r[6]]], [[[-6/7, -1/7, 4/7, 2/7], r[7]], [[-5/7, -5/7, 6/7, -6/7], 986/225-2057/1050*r[7]], [[-4/7, -3/7, 6/7, 2/7], 28/39+11/39*r[7]], [[-3/7, -\
3/7, 5/7, -5/7], -(8918-7007*r[7])/(-385+1815*r[7])]], [[[-5/7, -3/7, -4/7, -2/7], r[8]], [[-5/7, -3/7, -4/7, 5/7], 3+2/65*r[8]], [[-5/7, -3/7, 3/7, -2/7], 15/11-4/
143*r[8]], [[-5/7, -3/7, 3/7, 5/7], 27/11+8/715*r[8]], [[-5/7, -1/7, -4/7, -4/7], r[8]], [[-5/7, -1/7, -4/7, 3/7], 13/5+4/25*r[8]], [[-5/7, -1/7, 3/7, -4/7], 13/9-4
/45*r[8]], [[-5/7, -1/7, 3/7, 3/7], 91/45+8/225*r[8]], [[-5/7, 4/7, -4/7, -2/7], 13/5+4/25*r[8]], [[-5/7, 4/7, -4/7, 5/7], -21/5-4/325*r[8]], [[-5/7, 4/7, 3/7, -2/7
], 9/5-4/325*r[8]], [[-5/7, 4/7, 3/7, 5/7], -11/5-4/325*r[8]], [[-5/7, 6/7, -4/7, -4/7], 3+2/65*r[8]], [[-5/7, 6/7, -4/7, 3/7], -21/5-4/325*r[8]], [[-5/7, 6/7, 3/7,
-4/7], 9/5-4/325*r[8]], [[-5/7, 6/7, 3/7, 3/7], -6/5-4/325*r[8]], [[-4/7, -4/7, -3/7, -2/7], 12/13*r[8]], [[-4/7, -4/7, -3/7, 5/7], 5/2+1/39*r[8]], [[-4/7, -4/7, 2/
7, -1/7], 3/2-1/13*r[8]], [[-4/7, -4/7, 2/7, 6/7], 11/4+1/78*r[8]], [[-4/7, -4/7, 4/7, -2/7], 5/4-1/52*r[8]], [[-4/7, -4/7, 4/7, 5/7], 2+1/130*r[8]], [[-4/7, -2/7,
2/7, -3/7], 13/8-1/6*r[8]], [[-4/7, -2/7, 2/7, 4/7], 143/64+3/80*r[8]], [[-4/7, -1/7, -3/7, -5/7], 12/13*r[8]], [[-4/7, -1/7, -3/7, 2/7], 2+8/39*r[8]], [[-4/7, -1/7
, 4/7, -5/7], 4/3-16/195*r[8]], [[-4/7, -1/7, 4/7, 2/7], 19/12+1/39*r[8]], [[-4/7, 3/7, -3/7, -2/7], 2+8/39*r[8]], [[-4/7, 3/7, -3/7, 5/7], -25/3-4/117*r[8]], [[-4/
7, 3/7, 2/7, -1/7], 2-8/195*r[8]], [[-4/7, 3/7, 2/7, 6/7], -19/6-2/117*r[8]], [[-4/7, 3/7, 4/7, -2/7], 3/2-2/195*r[8]], [[-4/7, 3/7, 4/7, 5/7], -22/5-8/325*r[8]], [
[-4/7, 5/7, 2/7, -3/7], 2-8/195*r[8]], [[-4/7, 5/7, 2/7, 4/7], -15/8-3/130*r[8]], [[-4/7, 6/7, -3/7, -5/7], 5/2+1/39*r[8]], [[-4/7, 6/7, -3/7, 2/7], -25/3-4/117*r[8
]], [[-4/7, 6/7, 4/7, -5/7], 3/2-2/195*r[8]], [[-4/7, 6/7, 4/7, 2/7], -35/16-1/52*r[8]], [[-3/7, -5/7, 3/7, -1/7], 15/11-8/143*r[8]], [[-3/7, -5/7, 3/7, 6/7], 25/11
+4/429*r[8]], [[-3/7, -2/7, 3/7, -4/7], 3/2-2/13*r[8]], [[-3/7, -2/7, 3/7, 3/7], 7/4+1/39*r[8]], [[-3/7, 2/7, 3/7, -1/7], 5/3-4/117*r[8]], [[-3/7, 2/7, 3/7, 6/7], -\
19/3-4/117*r[8]], [[-3/7, 5/7, 3/7, -4/7], 5/3-4/117*r[8]], [[-3/7, 5/7, 3/7, 3/7], -10/3-4/117*r[8]], [[-2/7, -4/7, -1/7, -4/7], 4/5*r[8]], [[-2/7, -4/7, -1/7, 3/7
], 13/8+1/10*r[8]], [[-2/7, -4/7, 6/7, -4/7], 13/12-1/60*r[8]], [[-2/7, -4/7, 6/7, 3/7], 143/120+1/150*r[8]], [[-2/7, -3/7, -1/7, -5/7], 4/5*r[8]], [[-2/7, -3/7, -1
/7, 2/7], 3/2+2/13*r[8]], [[-2/7, -3/7, 6/7, -5/7], 12/11-16/715*r[8]], [[-2/7, -3/7, 6/7, 2/7], 51/44+1/143*r[8]], [[-2/7, 3/7, -1/7, -4/7], 3/2+2/13*r[8]], [[-2/7
, 3/7, -1/7, 3/7], 15/2+2/13*r[8]], [[-2/7, 3/7, 6/7, -4/7], 9/8-1/130*r[8]], [[-2/7, 3/7, 6/7, 3/7], 12/5+8/325*r[8]], [[-2/7, 4/7, -1/7, -5/7], 13/8+1/10*r[8]], [
[-2/7, 4/7, -1/7, 2/7], 15/2+2/13*r[8]], [[-2/7, 4/7, 6/7, -5/7], 9/8-1/130*r[8]], [[-2/7, 4/7, 6/7, 2/7], 35/16+1/52*r[8]], [[-1/7, -5/7, 5/7, -3/7], 13/11-8/165*r
[8]], [[-1/7, -5/7, 5/7, 4/7], 299/220+3/275*r[8]], [[-1/7, -4/7, 5/7, -4/7], 6/5-4/65*r[8]], [[-1/7, -4/7, 5/7, 3/7], 13/10+2/195*r[8]], [[-1/7, 2/7, 5/7, -3/7], 5
/4-1/39*r[8]], [[-1/7, 2/7, 5/7, 4/7], 15/4+3/65*r[8]], [[-1/7, 3/7, 5/7, -4/7], 5/4-1/39*r[8]], [[-1/7, 3/7, 5/7, 3/7], 10/3+4/117*r[8]], [[2/7, -3/7, 3/7, -2/7],
1+4/39*r[8]], [[2/7, -3/7, 3/7, 5/7], 5/3+4/585*r[8]], [[2/7, -1/7, 3/7, -4/7], 1+4/39*r[8]], [[2/7, -1/7, 3/7, 3/7], 3/2+2/65*r[8]], [[2/7, 4/7, 3/7, -2/7], 3/2+2/
65*r[8]], [[2/7, 4/7, 3/7, 5/7], -3/4-1/195*r[8]], [[2/7, 6/7, 3/7, -4/7], 5/3+4/585*r[8]], [[2/7, 6/7, 3/7, 3/7], -3/4-1/195*r[8]], [[3/7, -4/7, 4/7, -2/7], 1+4/65
*r[8]], [[3/7, -4/7, 4/7, 5/7], 7/5+4/975*r[8]], [[3/7, -1/7, 4/7, -5/7], 1+4/65*r[8]], [[3/7, -1/7, 4/7, 2/7], 5/4+1/39*r[8]], [[3/7, 3/7, 4/7, -2/7], 5/4+1/39*r[8
]], [[3/7, 3/7, 4/7, 5/7], -3/2-2/195*r[8]], [[3/7, 6/7, 4/7, -5/7], 7/5+4/975*r[8]], [[3/7, 6/7, 4/7, 2/7], -3/2-2/195*r[8]], [[5/7, -4/7, 6/7, -4/7], 1+2/195*r[8]
], [[5/7, -4/7, 6/7, 3/7], 21/20+1/325*r[8]], [[5/7, -3/7, 6/7, -5/7], 1+2/195*r[8]], [[5/7, -3/7, 6/7, 2/7], 25/24+1/234*r[8]], [[5/7, 3/7, 6/7, -4/7], 25/24+1/234
*r[8]], [[5/7, 3/7, 6/7, 3/7], 3/2+2/195*r[8]], [[5/7, 4/7, 6/7, -5/7], 21/20+1/325*r[8]], [[5/7, 4/7, 6/7, 2/7], 3/2+2/195*r[8]]]]]:
end:



#WadimHW(): A data-base of conjectured 3F2 identities (many of them probably equivalent via some transformation
#
WadimHW:=proc():
[[[[0, 0, 1/2, 0], 2*ln(2)], [[0, 0, 1/2, -1/2], 4-4*ln(2)], [[0, -1/2, 1/2, -1/2], -3+6*ln(2)], [[0, 1/2, 1/2, -1/2], 2*ln(2)], [[-1/2, 0, 0, 0], 6-6*ln(2)], [[-1/
2, -1/2, 0, 0], -4+8*ln(2)], [[-1/2, -1/2, 0, -1/2], 18-24*ln(2)], [[-1/2, 1/2, 0, 0], 4*ln(2)], [[-1/2, 1/2, 0, -1/2], 6-6*ln(2)]], [[[0, 0, 1/3, -2/3], -6+4/3*Pi*
3^(1/2)], [[0, 0, 1/3, 1/3], 1/3*Pi*3^(1/2)], [[0, 0, 2/3, -1/3], 3-1/3*Pi*3^(1/2)], [[0, 0, 2/3, 2/3], 3/2+1/6*Pi*3^(1/2)], [[0, -2/3, 2/3, -2/3], 45/8-5/6*Pi*3^(1
/2)], [[0, -1/3, 1/3, -1/3], -6+4/3*Pi*3^(1/2)], [[0, 1/3, 2/3, -2/3], 3-1/3*Pi*3^(1/2)], [[0, 2/3, 1/3, -1/3], 1/3*Pi*3^(1/2)], [[-2/3, 0, -1/3, 0], -10+20/9*Pi*3^
(1/2)], [[-2/3, 0, 2/3, 0], 5/8+5/36*Pi*3^(1/2)], [[-2/3, -2/3, 0, 0], 63/8-7/6*Pi*3^(1/2)], [[-2/3, -2/3, -1/3, -1/3], -245/6+70/9*Pi*3^(1/2)], [[-2/3, -1/3, 0, -1
/3], 16-8/3*Pi*3^(1/2)], [[-2/3, -1/3, 0, 2/3], 1+1/3*Pi*3^(1/2)], [[-2/3, 1/3, 0, 0], 6-2/3*Pi*3^(1/2)], [[-2/3, 1/3, -1/3, -1/3], -10+20/9*Pi*3^(1/2)], [[-2/3, 2/
3, 0, -1/3], 6-2/3*Pi*3^(1/2)], [[-2/3, 2/3, 0, 2/3], -3/2-1/6*Pi*3^(1/2)], [[-1/3, 0, 1/3, 0], 4-4/9*Pi*3^(1/2)], [[-1/3, -2/3, 0, -2/3], -35+20/3*Pi*3^(1/2)], [[-\
1/3, -2/3, 0, 1/3], -2+2/3*Pi*3^(1/2)], [[-1/3, -1/3, 0, 0], -15/2+5/3*Pi*3^(1/2)], [[-1/3, -1/3, 1/3, -2/3], 40/3-20/9*Pi*3^(1/2)], [[-1/3, 1/3, 0, -2/3], -15/2+5/
3*Pi*3^(1/2)], [[-1/3, 1/3, 0, 1/3], 2/3*Pi*3^(1/2)], [[-1/3, 2/3, 0, 0], 2/3*Pi*3^(1/2)], [[-1/3, 2/3, 1/3, -2/3], 4-4/9*Pi*3^(1/2)], [[1/3, 0, 2/3, 0], 2/9*Pi*3^(
1/2)], [[1/3, -2/3, 2/3, -1/3], -4/3+4/9*Pi*3^(1/2)], [[1/3, 1/3, 2/3, -1/3], 2/9*Pi*3^(1/2)]], [[[-3/4, -3/4, -1/4, -3/4], -(48+64*2^(1/2))/(-51-36*2^(1/2))], [[-3
/4, -3/4, -1/4, -1/4], -(4+12*2^(1/2))/(-7-5*2^(1/2))], [[-3/4, -3/4, -1/4, 1/4], -(8+16*2^(1/2))/(-9-6*2^(1/2))], [[-3/4, -3/4, -1/4, 3/4], -(12+4*2^(1/2))/(-3-3*2
^(1/2))], [[-3/4, -3/4, 3/4, -3/4], -(8-56*2^(1/2))/(27+27*2^(1/2))], [[-3/4, -3/4, 3/4, -1/4], 76/27-32/27*2^(1/2)], [[-3/4, -3/4, 3/4, 1/4], 8/9*2^(1/2)], [[-3/4,
-3/4, 3/4, 3/4], -20/(27-27*2^(1/2))], [[-3/4, -1/2, -3/4, -1/4], -(-24-24*2^(1/2))/(17+12*2^(1/2))], [[-3/4, -1/2, -3/4, 3/4], -(6+2^(1/2))/(-1-2^(1/2))], [[-3/4,
-1/2, 1/4, -1/4], -(-3-6*2^(1/2))/(4+3*2^(1/2))], [[-3/4, -1/2, 1/4, 3/4], 15/8*2^(1/2)], [[-3/4, -1/4, -3/4, 1/2], -(40+80*2^(1/2))/(-27-18*2^(1/2))], [[-3/4, -1/4
, -1/4, -1/4], -(16-32*2^(1/2))/(9+6*2^(1/2))], [[-3/4, -1/4, -1/4, 3/4], -(16+48*2^(1/2))/(-9-9*2^(1/2))], [[-3/4, -1/4, 1/4, -1/2], 8*2^(1/2)/(4+3*2^(1/2))], [[-3
/4, -1/4, 1/4, 1/2], -(-72-8*2^(1/2))/(15+15*2^(1/2))], [[-3/4, -1/4, 3/4, -1/4], -1/126*(-128+16*2^(1/2))*2^(1/2)], [[-3/4, -1/4, 3/4, 3/4], -(112+192*2^(1/2))/(-\
63-63*2^(1/2))], [[-3/4, 1/2, -3/4, 3/4], -20/(-9+9*2^(1/2))], [[-3/4, 1/2, 1/4, -1/4], 10*2^(1/2)/(3+3*2^(1/2))], [[-3/4, 1/2, 1/4, 3/4], -5/2*2^(1/2)], [[-3/4, 1/
4, -1/4, -3/4], -(16-32*2^(1/2))/(9+6*2^(1/2))], [[-3/4, 1/4, -1/4, -1/4], 12/(3+2*2^(1/2))], [[-3/4, 1/4, -1/4, 1/4], -(8+8*2^(1/2))/(-3-2*2^(1/2))], [[-3/4, 1/4,
3/4, -3/4], -(-8-24*2^(1/2))/(15+15*2^(1/2))], [[-3/4, 1/4, 3/4, -1/4], -(4+8*2^(1/2))/(-5-5*2^(1/2))], [[-3/4, 1/4, 3/4, 1/4], -(12+6*2^(1/2))/(-5-5*2^(1/2))], [[-\
3/4, 3/4, -1/4, -1/4], -(8+8*2^(1/2))/(-3-2*2^(1/2))], [[-3/4, 3/4, -1/4, 3/4], -(4+6*2^(1/2))/(3+3*2^(1/2))], [[-3/4, 3/4, 1/4, -1/2], 10*2^(1/2)/(3+3*2^(1/2))], [
[-3/4, 3/4, 1/4, 1/2], -(12+4*2^(1/2))/(3+3*2^(1/2))], [[-3/4, 3/4, 3/4, -1/4], 8/9+4/9*2^(1/2)], [[-3/4, 3/4, 3/4, 3/4], -(-2+3*2^(1/2))/(9+9*2^(1/2))], [[-1/2, -3
/4, -1/2, -1/4], 21/(7+5*2^(1/2))], [[-1/2, -3/4, -1/2, 3/4], -(3+9*2^(1/2))/(-4-2*2^(1/2))], [[-1/2, -3/4, 1/2, -1/4], -(9-21*2^(1/2))/(7+7*2^(1/2))], [[-1/2, -3/4
, 1/2, 3/4], 1/14*(15+3*2^(1/2))*2^(1/2)], [[-1/2, -1/4, -1/2, 1/4], 35*2^(1/2)/(12+8*2^(1/2))], [[-1/2, -1/4, 1/2, -3/4], 7*2^(1/2)/(4+3*2^(1/2))], [[-1/2, -1/4, 1
/2, 1/4], -(-7-14*2^(1/2))/(10+5*2^(1/2))], [[-1/2, 1/4, 1/2, -1/4], 5*2^(1/2)/(2+2*2^(1/2))], [[-1/2, 3/4, 1/2, -3/4], 5*2^(1/2)/(2+2*2^(1/2))], [[-1/4, -3/4, -1/4
, -1/2], -(56+56*2^(1/2))/(-51-36*2^(1/2))], [[-1/4, -3/4, -1/4, 1/2], -(8+16*2^(1/2))/(-9-6*2^(1/2))], [[-1/4, -3/4, 1/4, -3/4], -(16+48*2^(1/2))/(-35-25*2^(1/2))]
, [[-1/4, -3/4, 1/4, 1/4], -(16-48*2^(1/2))/(15+15*2^(1/2))], [[-1/4, -3/4, 3/4, -1/2], -(24-56*2^(1/2))/(21+21*2^(1/2))], [[-1/4, -3/4, 3/4, 1/2], -8/63+64/63*2^(1
/2)], [[-1/4, -1/2, -1/4, 1/4], 7*2^(1/2)/(3+2*2^(1/2))], [[-1/4, -1/2, 3/4, -3/4], -(7+14*2^(1/2))/(-12-9*2^(1/2))], [[-1/4, -1/2, 3/4, 1/4], 7/8*2^(1/2)], [[-1/4,
-1/4, 1/4, -3/4], -(4-8*2^(1/2))/(3+2*2^(1/2))], [[-1/4, -1/4, 1/4, -1/4], 8*2^(1/2)/(4+3*2^(1/2))], [[-1/4, -1/4, 1/4, 1/4], -(-4-4*2^(1/2))/(3+2*2^(1/2))], [[-1/4
, -1/4, 1/4, 3/4], 2*2^(1/2)], [[-1/4, 1/2, 3/4, -3/4], 2*2^(1/2)/(1+2^(1/2))], [[-1/4, 1/2, 3/4, 1/4], 3/2*2^(1/2)], [[-1/4, 1/4, -1/4, 1/2], 12/(1+2^(1/2))], [[-1
/4, 1/4, 1/4, -3/4], 8*2^(1/2)/(4+3*2^(1/2))], [[-1/4, 1/4, 1/4, 1/4], 4*2^(1/2)/(1+2^(1/2))], [[-1/4, 1/4, 3/4, -1/2], 2*2^(1/2)/(1+2^(1/2))], [[-1/4, 1/4, 3/4, 1/
2], -(12+4*2^(1/2))/(-3-3*2^(1/2))], [[-1/4, 3/4, 1/4, -3/4], -(-4-4*2^(1/2))/(3+2*2^(1/2))], [[-1/4, 3/4, 1/4, -1/4], 4*2^(1/2)/(1+2^(1/2))], [[-1/4, 3/4, 1/4, 3/4
], -1/2*2^(1/2)], [[1/2, -3/4, 1/2, -1/4], -(-15+5*2^(1/2))/(3+3*2^(1/2))], [[1/2, -3/4, 1/2, 3/4], 5/9+5/9*2^(1/2)], [[1/2, -1/4, 1/2, 1/4], 3*2^(1/2)/(2+2^(1/2))]
, [[1/4, -3/4, 3/4, -3/4], -(-8-16*2^(1/2))/(15+10*2^(1/2))], [[1/4, -3/4, 3/4, -1/4], -(4-12*2^(1/2))/(5+5*2^(1/2))], [[1/4, -3/4, 3/4, 1/4], 4/5*2^(1/2)], [[1/4,
-3/4, 3/4, 3/4], 4/5+2/5*2^(1/2)], [[1/4, -1/2, 1/4, -1/4], 5*2^(1/2)/(3+2*2^(1/2))], [[1/4, -1/2, 1/4, 3/4], 5/4*2^(1/2)], [[1/4, -1/4, 1/4, 1/2], -(-4-4*2^(1/2))/
(3+2*2^(1/2))], [[1/4, -1/4, 3/4, -1/4], 8/(3+3*2^(1/2))], [[1/4, -1/4, 3/4, 3/4], -(-4-6*2^(1/2))/(3+3*2^(1/2))], [[1/4, 1/2, 1/4, 3/4], -2^(1/2)], [[1/4, 1/4, 3/4
, -3/4], 8/(3+3*2^(1/2))], [[1/4, 1/4, 3/4, -1/4], 2*2^(1/2)/(1+2^(1/2))], [[1/4, 1/4, 3/4, 1/4], 2^(1/2)], [[1/4, 3/4, 3/4, -1/4], 2^(1/2)], [[1/4, 3/4, 3/4, 3/4],
1/4*2^(1/2)], [[3/4, -3/4, 3/4, -1/2], -4/3+5/3*2^(1/2)], [[3/4, -3/4, 3/4, 1/2], 4/9+4/9*2^(1/2)], [[3/4, -1/2, 3/4, 1/4], 3/4*2^(1/2)], [[3/4, 1/4, 3/4, 1/2], 2^(
1/2)]], [[[-4/5, -4/5, -2/5, -3/5], -(130-195*5^(1/2))/(108+48*5^(1/2))], [[-4/5, -4/5, -2/5, 2/5], 65/(16+8*5^(1/2))], [[-4/5, -4/5, -1/5, -1/5], -(13-65*5^(1/2))/
(44+20*5^(1/2))], [[-4/5, -4/5, -1/5, 4/5], -26/7+39/14*5^(1/2)], [[-4/5, -4/5, 3/5, -3/5], -(-65-260*5^(1/2))/(264+132*5^(1/2))], [[-4/5, -4/5, 3/5, 2/5], -13/11+
13/11*5^(1/2)], [[-4/5, -4/5, 4/5, -1/5], -(117-143*5^(1/2))/(56+56*5^(1/2))], [[-4/5, -4/5, 4/5, 4/5], 13/16*5^(1/2)], [[-4/5, -3/5, -1/5, -2/5], -(22+2*5^(1/2))/(
-9-4*5^(1/2))], [[-4/5, -3/5, -1/5, 3/5], -(-1-5*5^(1/2))/(3+5^(1/2))], [[-4/5, -3/5, 4/5, -2/5], -(12-23*5^(1/2))/(11+11*5^(1/2))], [[-4/5, -3/5, 4/5, 3/5], 1/11+7
/11*5^(1/2)], [[-4/5, -2/5, -3/5, -2/5], -(55-55*5^(1/2))/(21+9*5^(1/2))], [[-4/5, -2/5, -3/5, 3/5], 55/(6+6*5^(1/2))], [[-4/5, -2/5, 2/5, -2/5], 55*5^(1/2)/(40+24*
5^(1/2))], [[-4/5, -2/5, 2/5, 3/5], -11/(4-4*5^(1/2))], [[-4/5, -1/5, -3/5, -3/5], -(55-55*5^(1/2))/(21+9*5^(1/2))], [[-4/5, -1/5, -3/5, 2/5], -(5-15*5^(1/2))/(6+2*
5^(1/2))], [[-4/5, -1/5, -2/5, -1/5], -(20-20*5^(1/2))/(7+3*5^(1/2))], [[-4/5, -1/5, -2/5, 4/5], -(40+5*5^(1/2))/(-6-2*5^(1/2))], [[-4/5, -1/5, 2/5, -3/5], -(-5-15*
5^(1/2))/(14+7*5^(1/2))], [[-4/5, -1/5, 2/5, 2/5], -(-80-5*5^(1/2))/(14+14*5^(1/2))], [[-4/5, -1/5, 3/5, -1/5], -(5+10*5^(1/2))/(-12-4*5^(1/2))], [[-4/5, -1/5, 3/5,
4/5], -(5-45*5^(1/2))/(8+8*5^(1/2))], [[-4/5, 1/5, -2/5, -3/5], -(20-20*5^(1/2))/(7+3*5^(1/2))], [[-4/5, 1/5, -2/5, 2/5], 10*5^(1/2)/(3+5^(1/2))], [[-4/5, 1/5, -1/5
, -1/5], -(7-7*5^(1/2))/(2+5^(1/2))], [[-4/5, 1/5, 3/5, -3/5], -(5+5*5^(1/2))/(-6-3*5^(1/2))], [[-4/5, 1/5, 3/5, 2/5], -2+2*5^(1/2)], [[-4/5, 1/5, 4/5, -1/5], -(8-\
16*5^(1/2))/(7+7*5^(1/2))], [[-4/5, 2/5, -1/5, -2/5], -(7-7*5^(1/2))/(2+5^(1/2))], [[-4/5, 2/5, 4/5, -2/5], 7*5^(1/2)/(4+4*5^(1/2))], [[-4/5, 3/5, -3/5, -2/5], -(5-\
15*5^(1/2))/(6+2*5^(1/2))], [[-4/5, 3/5, -3/5, 3/5], -5/(3-5^(1/2))], [[-4/5, 3/5, 2/5, -2/5], 5*5^(1/2)/(2+2*5^(1/2))], [[-4/5, 3/5, 2/5, 3/5], -5^(1/2)-1], [[-4/5
, 4/5, -3/5, -3/5], 55/(6+6*5^(1/2))], [[-4/5, 4/5, -3/5, 2/5], -5/(3-5^(1/2))], [[-4/5, 4/5, -2/5, -1/5], 10*5^(1/2)/(3+5^(1/2))], [[-4/5, 4/5, -2/5, 4/5], -(15-10
*5^(1/2))/(4-4*5^(1/2))], [[-4/5, 4/5, 2/5, -3/5], 5*5^(1/2)/(2+2*5^(1/2))], [[-4/5, 4/5, 2/5, 2/5], -(20+5*5^(1/2))/(4+4*5^(1/2))], [[-4/5, 4/5, 3/5, -1/5], -5/(2-\
2*5^(1/2))], [[-3/5, -4/5, -1/5, -4/5], -(10-15*5^(1/2))/(9+4*5^(1/2))], [[-3/5, -4/5, -1/5, 1/5], 10*5^(1/2)/(7+3*5^(1/2))], [[-3/5, -4/5, 4/5, -4/5], -(-5-20*5^(1
/2))/(22+11*5^(1/2))], [[-3/5, -4/5, 4/5, 1/5], -21/22+21/22*5^(1/2)], [[-3/5, -3/5, -2/5, -2/5], 33/(11+5*5^(1/2))], [[-3/5, -3/5, -2/5, 3/5], -11/6+11/6*5^(1/2)],
[[-3/5, -3/5, 1/5, -1/5], 55*5^(1/2)/(42+21*5^(1/2))], [[-3/5, -3/5, 1/5, 4/5], 11/14+11/14*5^(1/2)], [[-3/5, -3/5, 3/5, -2/5], -(44-66*5^(1/2))/(27+27*5^(1/2))], [
[-3/5, -3/5, 3/5, 3/5], 22/27+11/27*5^(1/2)], [[-3/5, -2/5, -1/5, -1/5], 35/(11+5*5^(1/2))], [[-3/5, -2/5, -1/5, 4/5], -(15-35*5^(1/2))/(6+6*5^(1/2))], [[-3/5, -2/5
, 1/5, -2/5], -(5-10*5^(1/2))/(6+3*5^(1/2))], [[-3/5, -2/5, 1/5, 3/5], -5/6+25/18*5^(1/2)], [[-3/5, -2/5, 4/5, -1/5], -5/12+25/36*5^(1/2)], [[-3/5, -2/5, 4/5, 4/5],
-(-70-15*5^(1/2))/(27+9*5^(1/2))], [[-3/5, -1/5, -2/5, -4/5], 33/(11+5*5^(1/2))], [[-3/5, -1/5, -2/5, 1/5], 9/(2+5^(1/2))], [[-3/5, -1/5, 3/5, -4/5], -(-9-27*5^(1/2
))/(28+14*5^(1/2))], [[-3/5, -1/5, 3/5, 1/5], -3/7+6/7*5^(1/2)], [[-3/5, 1/5, -1/5, -4/5], 35/(11+5*5^(1/2))], [[-3/5, 1/5, -1/5, 1/5], 35/(6+3*5^(1/2))], [[-3/5, 1
/5, 4/5, -4/5], -(35-35*5^(1/2))/(12+12*5^(1/2))], [[-3/5, 1/5, 4/5, 1/5], -7/6+7/6*5^(1/2)], [[-3/5, 2/5, -2/5, -2/5], 9/(2+5^(1/2))], [[-3/5, 2/5, 1/5, -1/5], -(-\
5-5*5^(1/2))/(4+2*5^(1/2))], [[-3/5, 2/5, 1/5, 4/5], -3/2-3/2*5^(1/2)], [[-3/5, 2/5, 3/5, -2/5], 2*5^(1/2)/(5^(1/2)+1)], [[-3/5, 3/5, -1/5, -1/5], 35/(6+3*5^(1/2))]
, [[-3/5, 3/5, -1/5, 4/5], -10*5^(1/2)/(3+3*5^(1/2))], [[-3/5, 3/5, 1/5, -2/5], -(-5-5*5^(1/2))/(4+2*5^(1/2))], [[-3/5, 3/5, 1/5, 3/5], -5/3*5^(1/2)], [[-3/5, 3/5,
4/5, -1/5], -5/(3-3*5^(1/2))], [[-3/5, 4/5, -2/5, -4/5], -11/6+11/6*5^(1/2)], [[-3/5, 4/5, 3/5, -4/5], 2*5^(1/2)/(5^(1/2)+1)], [[-2/5, -4/5, 2/5, -1/5], -(44+11*5^(
1/2))/(-28-12*5^(1/2))], [[-2/5, -4/5, 2/5, 4/5], -(55+22*5^(1/2))/(-16-16*5^(1/2))], [[-2/5, -3/5, -1/5, -3/5], -(15-15*5^(1/2))/(7+3*5^(1/2))], [[-2/5, -3/5, -1/5
, 2/5], -(5-15*5^(1/2))/(9+3*5^(1/2))], [[-2/5, -3/5, 1/5, -4/5], -(55+5*5^(1/2))/(-27-12*5^(1/2))], [[-2/5, -3/5, 1/5, 1/5], -(5+15*5^(1/2))/(-12-6*5^(1/2))], [[-2
/5, -3/5, 4/5, -3/5], -(45+35*5^(1/2))/(-54-27*5^(1/2))], [[-2/5, -3/5, 4/5, 2/5], -(5-10*5^(1/2))/(54-18*5^(1/2))], [[-2/5, -2/5, -1/5, -4/5], -(15-15*5^(1/2))/(7+
3*5^(1/2))], [[-2/5, -2/5, -1/5, 1/5], 15/(4+2*5^(1/2))], [[-2/5, -2/5, 2/5, -3/5], -(3-6*5^(1/2))/(4+2*5^(1/2))], [[-2/5, -2/5, 2/5, 2/5], 3/4*5^(1/2)], [[-2/5, -2
/5, 4/5, -4/5], 45/(24+8*5^(1/2))], [[-2/5, -2/5, 4/5, 1/5], -3/(2-2*5^(1/2))], [[-2/5, -1/5, 1/5, -1/5], -(5-5*5^(1/2))/(2+5^(1/2))], [[-2/5, -1/5, 1/5, 4/5], -3+3
*5^(1/2)], [[-2/5, 1/5, 2/5, -1/5], -(-2-2*5^(1/2))/(2+5^(1/2))], [[-2/5, 2/5, -1/5, -3/5], 15/(4+2*5^(1/2))], [[-2/5, 2/5, -1/5, 2/5], -5+5*5^(1/2)], [[-2/5, 2/5,
1/5, -4/5], -(5-5*5^(1/2))/(2+5^(1/2))], [[-2/5, 2/5, 1/5, 1/5], 15*5^(1/2)/(5+3*5^(1/2))], [[-2/5, 2/5, 4/5, -3/5], 5*5^(1/2)/(3+3*5^(1/2))], [[-2/5, 2/5, 4/5, 2/5
], -(20+5*5^(1/2))/(-4-4*5^(1/2))], [[-2/5, 3/5, -1/5, -4/5], -(5-15*5^(1/2))/(9+3*5^(1/2))], [[-2/5, 3/5, -1/5, 1/5], -5+5*5^(1/2)], [[-2/5, 3/5, 2/5, -3/5], -(-2-\
2*5^(1/2))/(2+5^(1/2))], [[-2/5, 3/5, 4/5, -4/5], 5*5^(1/2)/(3+3*5^(1/2))], [[-2/5, 3/5, 4/5, 1/5], 2/3+2/3*5^(1/2)], [[-2/5, 4/5, 1/5, -1/5], 15*5^(1/2)/(5+3*5^(1/
2))], [[-2/5, 4/5, 1/5, 4/5], -2/(5^(1/2)+1)], [[-1/5, -4/5, 2/5, -4/5], -(5-25*5^(1/2))/(22+10*5^(1/2))], [[-1/5, -4/5, 2/5, 1/5], -(-60+5*5^(1/2))/(21+7*5^(1/2))]
, [[-1/5, -4/5, 3/5, -2/5], -(-5-15*5^(1/2))/(16+8*5^(1/2))], [[-1/5, -4/5, 3/5, 3/5], -(-35-10*5^(1/2))/(12+12*5^(1/2))], [[-1/5, -3/5, 3/5, -3/5], 15*5^(1/2)/(14+
7*5^(1/2))], [[-1/5, -3/5, 3/5, 2/5], 3/7+3/7*5^(1/2)], [[-1/5, -2/5, 1/5, -3/5], -(7+7*5^(1/2))/(-9-4*5^(1/2))], [[-1/5, -2/5, 1/5, 2/5], -(-5-5^(1/2))/(2+5^(1/2))
], [[-1/5, -1/5, 1/5, -4/5], -(7+7*5^(1/2))/(-9-4*5^(1/2))], [[-1/5, -1/5, 1/5, 1/5], 7/(2+5^(1/2))], [[-1/5, -1/5, 2/5, -2/5], 35/(14+6*5^(1/2))], [[-1/5, -1/5, 2/
5, 3/5], 7*5^(1/2)/(5+5^(1/2))], [[-1/5, 1/5, 2/5, -4/5], 35/(14+6*5^(1/2))], [[-1/5, 1/5, 2/5, 1/5], -(-5-5*5^(1/2))/(4+2*5^(1/2))], [[-1/5, 1/5, 3/5, -2/5], -(5+5
*5^(1/2))/(-6-3*5^(1/2))], [[-1/5, 1/5, 3/5, 3/5], 5/3*5^(1/2)], [[-1/5, 2/5, 3/5, -3/5], -(5+5*5^(1/2))/(-6-3*5^(1/2))], [[-1/5, 2/5, 3/5, 2/5], 5^(1/2)+1], [[-1/5
, 3/5, 1/5, -3/5], 7/(2+5^(1/2))], [[-1/5, 4/5, 1/5, -4/5], -(-5-5^(1/2))/(2+5^(1/2))], [[-1/5, 4/5, 2/5, -2/5], -(-5-5*5^(1/2))/(4+2*5^(1/2))], [[-1/5, 4/5, 2/5, 3
/5], -2/(5^(1/2)+1)], [[1/5, -4/5, 3/5, -3/5], 20*5^(1/2)/(21+9*5^(1/2))], [[1/5, -4/5, 3/5, 2/5], 5^(1/2)-1], [[1/5, -4/5, 4/5, -1/5], -(7+11*5^(1/2))/(-14-7*5^(1/
2))], [[1/5, -4/5, 4/5, 4/5], 3/7+3/7*5^(1/2)], [[1/5, -3/5, 4/5, -2/5], -(21-14*5^(1/2))/(3+3*5^(1/2))], [[1/5, -3/5, 4/5, 3/5], 7/18+7/18*5^(1/2)], [[1/5, -2/5, 2
/5, -2/5], -(5-5*5^(1/2))/(3+5^(1/2))], [[1/5, -2/5, 2/5, 3/5], -2/(-5^(1/2)+1)], [[1/5, -1/5, 2/5, -3/5], -(5-5*5^(1/2))/(3+5^(1/2))], [[1/5, -1/5, 2/5, 2/5], 5/(5
^(1/2)+1)], [[1/5, -1/5, 3/5, -1/5], -(5-5*5^(1/2))/(3+5^(1/2))], [[1/5, -1/5, 3/5, 4/5], 10*5^(1/2)/(3+3*5^(1/2))], [[1/5, 1/5, 3/5, -3/5], -(5-5*5^(1/2))/(3+5^(1/
2))], [[1/5, 1/5, 3/5, 2/5], -(3+3*5^(1/2))/(-3-5^(1/2))], [[1/5, 1/5, 4/5, -1/5], 6/(3+5^(1/2))], [[1/5, 2/5, 4/5, -2/5], 6/(3+5^(1/2))], [[1/5, 3/5, 2/5, -2/5], 5
/(5^(1/2)+1)], [[1/5, 3/5, 2/5, 3/5], -2/(5^(1/2)-1)], [[1/5, 4/5, 2/5, -3/5], -2/(-5^(1/2)+1)], [[1/5, 4/5, 2/5, 2/5], -2/(5^(1/2)-1)], [[1/5, 4/5, 3/5, -1/5], -(3
+3*5^(1/2))/(-3-5^(1/2))], [[1/5, 4/5, 3/5, 4/5], 2/(3+3*5^(1/2))], [[2/5, -4/5, 4/5, -4/5], 35*5^(1/2)/(40+16*5^(1/2))], [[2/5, -4/5, 4/5, 1/5], 7/(2+2*5^(1/2))],
[[2/5, -3/5, 3/5, -2/5], -(1-3*5^(1/2))/(3+5^(1/2))], [[2/5, -3/5, 3/5, 3/5], -1/(5^(1/2)-3)], [[2/5, -2/5, 4/5, -1/5], 5*5^(1/2)/(6+2*5^(1/2))], [[2/5, -2/5, 4/5,
4/5], -(-15+10*5^(1/2))/(4-4*5^(1/2))], [[2/5, -1/5, 3/5, -4/5], -(1-3*5^(1/2))/(3+5^(1/2))], [[2/5, -1/5, 3/5, 1/5], 5^(1/2)-1], [[2/5, 1/5, 4/5, -4/5], 5*5^(1/2)/
(6+2*5^(1/2))], [[2/5, 1/5, 4/5, 1/5], 5^(1/2)-1], [[2/5, 2/5, 3/5, -2/5], 5^(1/2)-1], [[2/5, 3/5, 4/5, -1/5], 5^(1/2)-1], [[2/5, 3/5, 4/5, 4/5], 1/(5^(1/2)+1)], [[
2/5, 4/5, 3/5, -4/5], -1/(5^(1/2)-3)], [[3/5, -3/5, 4/5, -3/5], -(5+5*5^(1/2))/(-9-3*5^(1/2))], [[3/5, -3/5, 4/5, 2/5], -(10+5*5^(1/2))/(-6-6*5^(1/2))], [[3/5, -2/5
, 4/5, -4/5], -(5+5*5^(1/2))/(-9-3*5^(1/2))], [[3/5, -2/5, 4/5, 1/5], -4/(3-3*5^(1/2))], [[3/5, 2/5, 4/5, -3/5], -4/(3-3*5^(1/2))], [[3/5, 2/5, 4/5, 2/5], -2/(-5^(1
/2)+1)], [[3/5, 3/5, 4/5, -4/5], -(10+5*5^(1/2))/(-6-6*5^(1/2))], [[3/5, 3/5, 4/5, 1/5], -2/(-5^(1/2)+1)]], [[[-5/6, -5/6, -1/2, -1/2], -(-1792+320*3^(1/2))/(420+
245*3^(1/2))], [[-5/6, -5/6, -1/2, 1/2], -(160-288*3^(1/2))/(105+35*3^(1/2))], [[-5/6, -5/6, -1/6, -1/6], -(-512-640*3^(1/2))/(540+315*3^(1/2))], [[-5/6, -5/6, -1/6
, 5/6], -1/135*(-256+32*3^(1/2))*3^(1/2)], [[-5/6, -5/6, 1/2, -1/2], -(-544-1728*3^(1/2))/(1365+910*3^(1/2))], [[-5/6, -5/6, 1/2, 1/2], 256/273*3^(1/2)], [[-5/6, -5
/6, 5/6, -1/6], -1/675*(512-544*3^(1/2))*3^(1/2)], [[-5/6, -5/6, 5/6, 5/6], 1/225*(128+64*3^(1/2))*3^(1/2)], [[-5/6, -1/2, -1/6, -1/2], -(16+52*3^(1/2))/(-36-21*3^(
1/2))], [[-5/6, -1/2, -1/6, 1/2], -1/45*(-128+40*3^(1/2))*3^(1/2)], [[-5/6, -1/2, 5/6, -1/2], -1/585*(256-360*3^(1/2))*3^(1/2)], [[-5/6, -1/2, 5/6, 1/2], 1/585*(224
+140*3^(1/2))*3^(1/2)], [[-5/6, -1/3, -1/2, -1/2], 13*3^(1/2)/(7+4*3^(1/2))], [[-5/6, -1/3, -1/2, 1/2], -(-65-78*3^(1/2))/(40+20*3^(1/2))], [[-5/6, -1/3, 1/2, -1/2]
, -(65-117*3^(1/2))/(40+40*3^(1/2))], [[-5/6, -1/3, 1/2, 1/2], 91/80*3^(1/2)], [[-5/6, -1/6, -1/2, -2/3], 13*3^(1/2)/(7+4*3^(1/2))], [[-5/6, -1/6, -1/2, -1/6], -(32
+8*3^(1/2))/(-12-7*3^(1/2))], [[-5/6, -1/6, -1/2, 1/3], 28*3^(1/2)/(10+5*3^(1/2))], [[-5/6, -1/6, -1/2, 5/6], -(80+64*3^(1/2))/(-15-10*3^(1/2))], [[-5/6, -1/6, 1/2,
-2/3], 14/(6+3*3^(1/2))], [[-5/6, -1/6, 1/2, -1/6], -(-80-32*3^(1/2))/(45+30*3^(1/2))], [[-5/6, -1/6, 1/2, 1/3], -28/15+32/15*3^(1/2)], [[-5/6, -1/6, 1/2, 5/6], -1/
45*(56-104*3^(1/2))*3^(1/2)], [[-5/6, 1/2, -1/6, -1/2], -(16-32*3^(1/2))/(9+6*3^(1/2))], [[-5/6, 1/2, 5/6, -1/2], -1/315*(64-160*3^(1/2))*3^(1/2)], [[-5/6, 1/6, -1/
2, -1/2], -(32+8*3^(1/2))/(-12-7*3^(1/2))], [[-5/6, 1/6, -1/2, 1/2], -1/3*(-16+4*3^(1/2))*3^(1/2)], [[-5/6, 1/6, -1/6, -1/6], -(16-32*3^(1/2))/(9+6*3^(1/2))], [[-5/
6, 1/6, 1/2, -1/2], -(12-52*3^(1/2))/(21+21*3^(1/2))], [[-5/6, 1/6, 1/2, 1/2], 40/21*3^(1/2)], [[-5/6, 1/6, 5/6, -1/6], -1/27*(16-20*3^(1/2))*3^(1/2)], [[-5/6, 2/3,
-1/2, -1/2], 28*3^(1/2)/(10+5*3^(1/2))], [[-5/6, 2/3, -1/2, 1/2], -(-14+14*3^(1/2))/(10-5*3^(1/2))], [[-5/6, 2/3, 1/2, -1/2], 14/5-7/10*3^(1/2)], [[-5/6, 2/3, 1/2,
1/2], -7/4*3^(1/2)], [[-5/6, 5/6, -1/2, -2/3], -(-65-78*3^(1/2))/(40+20*3^(1/2))], [[-5/6, 5/6, -1/2, -1/6], -1/3*(-16+4*3^(1/2))*3^(1/2)], [[-5/6, 5/6, -1/2, 1/3],
-(-14+14*3^(1/2))/(10-5*3^(1/2))], [[-5/6, 5/6, -1/2, 5/6], -1/15*(-2+8*3^(1/2))*3^(1/2)], [[-5/6, 5/6, 1/2, -2/3], 14/5-7/10*3^(1/2)], [[-5/6, 5/6, 1/2, -1/6], 1/
15*(8+8*3^(1/2))*3^(1/2)], [[-5/6, 5/6, 1/2, 1/3], -(20+14*3^(1/2))/(10+5*3^(1/2))], [[-5/6, 5/6, 1/2, 5/6], -1/30*(-7+8*3^(1/2))*3^(1/2)], [[-2/3, -1/2, -1/3, -1/2
], 143/(48+28*3^(1/2))], [[-2/3, -1/2, -1/3, 1/2], -(-65-78*3^(1/2))/(48+24*3^(1/2))], [[-2/3, -1/2, 2/3, -1/2], -1/66*(91-78*3^(1/2))*3^(1/2)], [[-2/3, -1/2, 2/3,
1/2], 13/33+91/132*3^(1/2)], [[-2/3, -1/6, -1/3, -5/6], 143/(48+28*3^(1/2))], [[-2/3, -1/6, -1/3, 1/6], -(77-154*3^(1/2))/(48+24*3^(1/2))], [[-2/3, -1/6, 2/3, -5/6]
, 77*3^(1/2)/(54+36*3^(1/2))], [[-2/3, -1/6, 2/3, 1/6], -11/9+55/36*3^(1/2)], [[-2/3, 1/2, -1/3, -1/2], -(77-154*3^(1/2))/(48+24*3^(1/2))], [[-2/3, 1/2, 2/3, -1/2],
7/3-7/12*3^(1/2)], [[-2/3, 5/6, -1/3, -5/6], -(-65-78*3^(1/2))/(48+24*3^(1/2))], [[-2/3, 5/6, 2/3, -5/6], 7/3-7/12*3^(1/2)], [[-1/2, -5/6, -1/6, -5/6], -(-224+40*3^
(1/2))/(60+35*3^(1/2))], [[-1/2, -5/6, -1/6, 1/6], -(-92+24*3^(1/2))/(15+10*3^(1/2))], [[-1/2, -5/6, 5/6, -5/6], -(784+500*3^(1/2))/(-780-455*3^(1/2))], [[-1/2, -5/
6, 5/6, 1/6], 128/195*3^(1/2)], [[-1/2, -2/3, 1/6, -1/6], -(-65-221*3^(1/2))/(160+96*3^(1/2))], [[-1/2, -2/3, 1/6, 5/6], 91/64*3^(1/2)], [[-1/2, -1/2, -1/6, -2/3],
11*3^(1/2)/(7+4*3^(1/2))], [[-1/2, -1/2, -1/6, -1/6], -(32-96*3^(1/2))/(45+25*3^(1/2))], [[-1/2, -1/2, -1/6, 1/3], 4*3^(1/2)/(2+3^(1/2))], [[-1/2, -1/2, -1/6, 5/6],
-1/15*(16-24*3^(1/2))*3^(1/2)], [[-1/2, -1/2, 1/6, -5/6], -(-32-104*3^(1/2))/(84+49*3^(1/2))], [[-1/2, -1/2, 1/6, -1/3], -(12+4*3^(1/2))/(-7-4*3^(1/2))], [[-1/2, -1
/2, 1/6, 1/6], -(16+32*3^(1/2))/(-21-14*3^(1/2))], [[-1/2, -1/2, 1/6, 2/3], -12/7+16/7*3^(1/2)], [[-1/2, -1/2, 5/6, -2/3], -(6-22*3^(1/2))/(11+11*3^(1/2))], [[-1/2,
-1/2, 5/6, -1/6], 1/165*(64+24*3^(1/2))*3^(1/2)], [[-1/2, -1/2, 5/6, 1/3], 12/55+32/55*3^(1/2)], [[-1/2, -1/2, 5/6, 5/6], -1/165*(56-144*3^(1/2))*3^(1/2)], [[-1/2,
-1/3, -1/6, -5/6], 11*3^(1/2)/(7+4*3^(1/2))], [[-1/2, -1/3, -1/6, 1/6], -(11-22*3^(1/2))/(8+4*3^(1/2))], [[-1/2, -1/3, 5/6, -5/6], -(55-99*3^(1/2))/(40+40*3^(1/2))]
, [[-1/2, -1/3, 5/6, 1/6], 11/16*3^(1/2)], [[-1/2, -1/6, 1/6, -1/6], -(4-8*3^(1/2))/(3+2*3^(1/2))], [[-1/2, -1/6, 1/6, 5/6], 8/3*3^(1/2)], [[-1/2, 1/2, -1/6, -2/3],
-(11-22*3^(1/2))/(8+4*3^(1/2))], [[-1/2, 1/2, -1/6, -1/6], 16*3^(1/2)/(6+3*3^(1/2))], [[-1/2, 1/2, -1/6, 1/3], -(-10-10*3^(1/2))/(2+3^(1/2))], [[-1/2, 1/2, -1/6, 5/
6], -1/3*(-2+4*3^(1/2))*3^(1/2)], [[-1/2, 1/2, 1/6, -5/6], -(4-8*3^(1/2))/(3+2*3^(1/2))], [[-1/2, 1/2, 1/6, -1/3], 7/(2+3^(1/2))], [[-1/2, 1/2, 1/6, 1/6], -(-8-8*3^
(1/2))/(3+2*3^(1/2))], [[-1/2, 1/2, 1/6, 2/3], -(8+6*3^(1/2))/(2+3^(1/2))], [[-1/2, 1/2, 5/6, -2/3], -(-1-3*3^(1/2))/(2+2*3^(1/2))], [[-1/2, 1/2, 5/6, -1/6], 1/15*(
4+4*3^(1/2))*3^(1/2)], [[-1/2, 1/2, 5/6, 1/3], -(-20-14*3^(1/2))/(10+5*3^(1/2))], [[-1/2, 1/2, 5/6, 5/6], -1/15*(-7+8*3^(1/2))*3^(1/2)], [[-1/2, 1/3, 1/6, -1/6], 7/
(2+3^(1/2))], [[-1/2, 1/3, 1/6, 5/6], -7/2*3^(1/2)], [[-1/2, 1/6, -1/6, -5/6], -(32-96*3^(1/2))/(45+25*3^(1/2))], [[-1/2, 1/6, -1/6, 1/6], 16*3^(1/2)/(6+3*3^(1/2))]
, [[-1/2, 1/6, 5/6, -5/6], -1/105*(128-112*3^(1/2))*3^(1/2)], [[-1/2, 1/6, 5/6, 1/6], 16/21*3^(1/2)], [[-1/2, 2/3, -1/6, -5/6], 4*3^(1/2)/(2+3^(1/2))], [[-1/2, 2/3,
-1/6, 1/6], -(-10-10*3^(1/2))/(2+3^(1/2))], [[-1/2, 2/3, 5/6, -5/6], -(-1-3*3^(1/2))/(2+2*3^(1/2))], [[-1/2, 2/3, 5/6, 1/6], 5/4*3^(1/2)], [[-1/2, 5/6, 1/6, -1/6],
-(-8-8*3^(1/2))/(3+2*3^(1/2))], [[-1/2, 5/6, 1/6, 5/6], -1/3*3^(1/2)], [[-1/3, -5/6, 1/3, -1/6], -(-65+13*3^(1/2))/(18+9*3^(1/2))], [[-1/3, -5/6, 1/3, 5/6], 13/45+
91/90*3^(1/2)], [[-1/3, -1/2, 1/3, -1/2], -(-11-11*3^(1/2))/(12+7*3^(1/2))], [[-1/3, -1/2, 1/3, 1/2], -11/21+55/42*3^(1/2)], [[-1/3, 1/2, 1/3, -1/2], -(35-35*3^(1/2
))/(6+6*3^(1/2))], [[-1/3, 1/6, 1/3, -1/6], -(35-35*3^(1/2))/(6+6*3^(1/2))], [[-1/6, -5/6, 1/2, -5/6], -(-128-160*3^(1/2))/(180+105*3^(1/2))], [[-1/6, -5/6, 1/2, -1
/3], -(-20+4*3^(1/2))/(6+3*3^(1/2))], [[-1/6, -5/6, 1/2, 1/6], -(-40-48*3^(1/2))/(45+30*3^(1/2))], [[-1/6, -5/6, 1/2, 2/3], -4/15+16/15*3^(1/2)], [[-1/6, -2/3, 1/2,
-1/2], -(-253+66*3^(1/2))/(64+32*3^(1/2))], [[-1/6, -2/3, 1/2, 1/2], 55/64*3^(1/2)], [[-1/6, -1/2, 1/6, -1/2], -(16-48*3^(1/2))/(27+15*3^(1/2))], [[-1/6, -1/2, 1/6,
1/2], -(-4-12*3^(1/2))/(9+3*3^(1/2))], [[-1/6, -1/6, 1/2, -1/2], -(16-32*3^(1/2))/(15+10*3^(1/2))], [[-1/6, -1/6, 1/2, 1/2], 16/15*3^(1/2)], [[-1/6, -1/6, 1/6, -5/6
], -(16-48*3^(1/2))/(27+15*3^(1/2))], [[-1/6, -1/6, 1/6, 1/6], -(32-32*3^(1/2))/(9+3*3^(1/2))], [[-1/6, 1/2, 1/6, -1/2], -(32-32*3^(1/2))/(9+3*3^(1/2))], [[-1/6, 1/
3, 1/2, -1/2], 5/(2+3^(1/2))], [[-1/6, 1/3, 1/2, 1/2], 5/2*3^(1/2)], [[-1/6, 1/6, 1/2, -5/6], -(16-32*3^(1/2))/(15+10*3^(1/2))], [[-1/6, 1/6, 1/2, -1/3], 5/(2+3^(1/
2))], [[-1/6, 1/6, 1/2, 1/6], -1/3*(4-4*3^(1/2))*3^(1/2)], [[-1/6, 1/6, 1/2, 2/3], -(-8-6*3^(1/2))/(2+3^(1/2))], [[-1/6, 5/6, 1/2, -1/2], -1/3*(4-4*3^(1/2))*3^(1/2)
], [[-1/6, 5/6, 1/2, 1/2], -1/3*3^(1/2)], [[-1/6, 5/6, 1/6, -5/6], -(-4-12*3^(1/2))/(9+3*3^(1/2))], [[1/2, -5/6, 5/6, -5/6], -(-176-32*3^(1/2))/(105+70*3^(1/2))], [
[1/2, -5/6, 5/6, 1/6], 64/105*3^(1/2)], [[1/2, -1/2, 5/6, -2/3], -2+7/4*3^(1/2)], [[1/2, -1/2, 5/6, -1/6], -1/15*(-16+4*3^(1/2))*3^(1/2)], [[1/2, -1/2, 5/6, 1/3], 1
/15*(6+2*3^(1/2))*3^(1/2)], [[1/2, -1/2, 5/6, 5/6], -1/15*(2-8*3^(1/2))*3^(1/2)], [[1/2, -1/3, 5/6, -5/6], -2+7/4*3^(1/2)], [[1/2, -1/3, 5/6, 1/6], 5/8*3^(1/2)], [[
1/2, 1/2, 5/6, -2/3], 5/8*3^(1/2)], [[1/2, 1/2, 5/6, -1/6], 2/3*3^(1/2)], [[1/2, 1/2, 5/6, 1/3], 3^(1/2)], [[1/2, 1/2, 5/6, 5/6], 1/6*3^(1/2)], [[1/2, 1/6, 5/6, -5/
6], -1/15*(-16+4*3^(1/2))*3^(1/2)], [[1/2, 1/6, 5/6, 1/6], 2/3*3^(1/2)], [[1/2, 2/3, 5/6, -5/6], 1/15*(6+2*3^(1/2))*3^(1/2)], [[1/2, 2/3, 5/6, 1/6], 3^(1/2)], [[1/3
, -1/2, 2/3, -1/2], 7/(3+2*3^(1/2))], [[1/3, -1/2, 2/3, 1/2], 7/15+7/15*3^(1/2)], [[1/3, -1/6, 2/3, -5/6], 7/(3+2*3^(1/2))], [[1/3, -1/6, 2/3, 1/6], -5/3+5/3*3^(1/2
)], [[1/3, 1/2, 2/3, -1/2], -5/3+5/3*3^(1/2)], [[1/3, 5/6, 2/3, -5/6], 7/15+7/15*3^(1/2)], [[1/6, -5/6, 1/2, -1/2], -(112+44*3^(1/2))/(-84-49*3^(1/2))], [[1/6, -5/6
, 1/2, 1/2], 16/21*3^(1/2)], [[1/6, -5/6, 5/6, -1/6], -(20+24*3^(1/2))/(-27-18*3^(1/2))], [[1/6, -5/6, 5/6, 5/6], 1/27*(8+8*3^(1/2))*3^(1/2)], [[1/6, -1/2, 5/6, -1/
2], -(-32-64*3^(1/2))/(63+42*3^(1/2))], [[1/6, -1/2, 5/6, 1/2], 1/63*(16+16*3^(1/2))*3^(1/2)], [[1/6, -1/3, 1/2, -1/2], -(28+21*3^(1/2))/(-28-16*3^(1/2))], [[1/6, -\
1/3, 1/2, 1/2], 7/8*3^(1/2)], [[1/6, -1/6, 1/2, -2/3], -(28+21*3^(1/2))/(-28-16*3^(1/2))], [[1/6, -1/6, 1/2, -1/6], 8/(3+2*3^(1/2))], [[1/6, -1/6, 1/2, 1/3], 4*3^(1
/2)/(3+3^(1/2))], [[1/6, -1/6, 1/2, 5/6], -1/3*(2-4*3^(1/2))*3^(1/2)], [[1/6, 1/2, 5/6, -1/2], -1/9*(8-8*3^(1/2))*3^(1/2)], [[1/6, 1/6, 1/2, -1/2], 8/(3+2*3^(1/2))]
, [[1/6, 1/6, 1/2, 1/2], 4/3*3^(1/2)], [[1/6, 1/6, 5/6, -1/6], -1/9*(8-8*3^(1/2))*3^(1/2)], [[1/6, 2/3, 1/2, -1/2], 4*3^(1/2)/(3+3^(1/2))], [[1/6, 2/3, 1/2, 1/2], -
3^(1/2)], [[1/6, 5/6, 1/2, -2/3], 7/8*3^(1/2)], [[1/6, 5/6, 1/2, -1/6], 4/3*3^(1/2)], [[1/6, 5/6, 1/2, 1/3], -3^(1/2)], [[1/6, 5/6, 1/2, 5/6], 1/12*3^(1/2)]], [[[-5
/6, -5/6, -1/3, -2/3], -1/10*(-1944+1536*2^(1/3))*2^(2/3)], [[-5/6, -5/6, -1/3, -1/6], -1/10*(648-522*2^(1/3))*2^(2/3)], [[-5/6, -5/6, -1/3, 1/3], -1/10*(-72+48*2^(
1/3))*2^(2/3)], [[-5/6, -5/6, -1/3, 5/6], -1/10*(-24+6*2^(1/3))*2^(2/3)], [[-5/6, -5/6, 2/3, -2/3], -1/70*(162-168*2^(1/3))*2^(2/3)], [[-5/6, -5/6, 2/3, -1/6], -1/
70*(144-156*2^(1/3))*2^(2/3)], [[-5/6, -5/6, 2/3, 1/3], 6/7*2^(2/3)], [[-5/6, -5/6, 2/3, 5/6], 1/70*(64+24*2^(1/3))*2^(2/3)], [[-5/6, -2/3, -1/3, -2/3], -261+1458/7
*2^(1/3)], [[-5/6, -2/3, -1/3, -1/3], -1/14*(1215-975*2^(1/3))*2^(2/3)], [[-5/6, -2/3, -1/3, 1/3], -9+243/28*2^(1/3)], [[-5/6, -2/3, -1/3, 2/3], -1/28*(-81+30*2^(1/
3))*2^(2/3)], [[-5/6, -2/3, 2/3, -2/3], 9/2-243/91*2^(1/3)], [[-5/6, -2/3, 2/3, -1/3], -1/182*(243-300*2^(1/3))*2^(2/3)], [[-5/6, -2/3, 2/3, 1/3], 405/364*2^(1/3)],
[[-5/6, -2/3, 2/3, 2/3], 1/104*(81+30*2^(1/3))*2^(2/3)], [[-5/6, -1/2, -2/3, -1/3], -210+168*2^(1/3)], [[-5/6, -1/2, -2/3, 2/3], -21/5+28/5*2^(1/3)], [[-5/6, -1/2,
1/3, -1/3], 42/5-28/5*2^(1/3)], [[-5/6, -1/2, 1/3, 2/3], 28/15*2^(1/3)], [[-5/6, -1/3, -2/3, -1/3], -1/10*(-1092+858*2^(1/3))*2^(2/3)], [[-5/6, -1/3, -2/3, 2/3], -1
/20*(-91+39*2^(1/3))*2^(2/3)], [[-5/6, -1/3, 1/3, -1/3], -1/90*(364-351*2^(1/3))*2^(2/3)], [[-5/6, -1/3, 1/3, 2/3], 91/54*2^(2/3)], [[-5/6, -1/6, -2/3, -2/3], -210+
168*2^(1/3)], [[-5/6, -1/6, -2/3, -1/2], -1/10*(-1092+858*2^(1/3))*2^(2/3)], [[-5/6, -1/6, -2/3, 1/2], -1/10*(-126+84*2^(1/3))*2^(2/3)], [[-5/6, -1/6, -2/3, 1/3], -\
168/5+144/5*2^(1/3)], [[-5/6, -1/6, -1/3, -1/3], -1/2*(-108+84*2^(1/3))*2^(2/3)], [[-5/6, -1/6, -1/3, -1/6], -297/5+243/5*2^(1/3)], [[-5/6, -1/6, -1/3, 2/3], -1/10*
(36-48*2^(1/3))*2^(2/3)], [[-5/6, -1/6, -1/3, 5/6], 63/5-27/5*2^(1/3)], [[-5/6, -1/6, 1/3, -2/3], 24-18*2^(1/3)], [[-5/6, -1/6, 1/3, -1/2], -1/20*(168-147*2^(1/3))*
2^(2/3)], [[-5/6, -1/6, 1/3, 1/2], -1/20*(-72+33*2^(1/3))*2^(2/3)], [[-5/6, -1/6, 1/3, 1/3], -24/5+27/5*2^(1/3)], [[-5/6, -1/6, 2/3, -1/3], -1/50*(-54+12*2^(1/3))*2
^(2/3)], [[-5/6, -1/6, 2/3, -1/6], -36/25+54/25*2^(1/3)], [[-5/6, -1/6, 2/3, 2/3], -1/50*(63-114*2^(1/3))*2^(2/3)], [[-5/6, -1/6, 2/3, 5/6], 198/25-72/25*2^(1/3)],
[[-5/6, 1/2, -2/3, -1/3], -168/5+144/5*2^(1/3)], [[-5/6, 1/2, -2/3, 2/3], -12/5-24/5*2^(1/3)], [[-5/6, 1/2, 1/3, -1/3], 24/5-12/5*2^(1/3)], [[-5/6, 1/2, 1/3, 2/3],
-4*2^(1/3)], [[-5/6, 1/3, -1/3, -2/3], -297/5+243/5*2^(1/3)], [[-5/6, 1/3, -1/3, -1/3], -1/2*(54-45*2^(1/3))*2^(2/3)], [[-5/6, 1/3, -1/3, 1/3], -27/5+81/10*2^(1/3)]
, [[-5/6, 1/3, 2/3, -2/3], 27/10-81/70*2^(1/3)], [[-5/6, 1/3, 2/3, -1/3], -1/70*(54-90*2^(1/3))*2^(2/3)], [[-5/6, 1/3, 2/3, 1/3], 27/14*2^(1/3)], [[-5/6, 1/6, -1/3,
-2/3], -1/2*(-108+84*2^(1/3))*2^(2/3)], [[-5/6, 1/6, -1/3, -1/6], -1/2*(54-45*2^(1/3))*2^(2/3)], [[-5/6, 1/6, -1/3, 1/3], -1/2*(-12+6*2^(1/3))*2^(2/3)], [[-5/6, 1/6
, 2/3, -2/3], -1/8*(9-12*2^(1/3))*2^(2/3)], [[-5/6, 1/6, 2/3, -1/6], -1/8*(12-15*2^(1/3))*2^(2/3)], [[-5/6, 1/6, 2/3, 1/3], 5/4*2^(2/3)], [[-5/6, 2/3, -2/3, -1/3],
-1/10*(-126+84*2^(1/3))*2^(2/3)], [[-5/6, 2/3, -2/3, 2/3], -1/20*(21+21*2^(1/3))*2^(2/3)], [[-5/6, 2/3, 1/3, -1/3], -1/10*(14-21*2^(1/3))*2^(2/3)], [[-5/6, 2/3, 1/3
, 2/3], -7/6*2^(2/3)], [[-5/6, 5/6, -2/3, -2/3], -21/5+28/5*2^(1/3)], [[-5/6, 5/6, -2/3, -1/2], -1/20*(-91+39*2^(1/3))*2^(2/3)], [[-5/6, 5/6, -2/3, 1/2], -1/20*(21+
21*2^(1/3))*2^(2/3)], [[-5/6, 5/6, -2/3, 1/3], -12/5-24/5*2^(1/3)], [[-5/6, 5/6, -1/3, -1/3], -1/2*(-12+6*2^(1/3))*2^(2/3)], [[-5/6, 5/6, -1/3, -1/6], -27/5+81/10*2
^(1/3)], [[-5/6, 5/6, -1/3, 2/3], -1/10*(-4+12*2^(1/3))*2^(2/3)], [[-5/6, 5/6, -1/3, 5/6], -9/5+9/20*2^(1/3)], [[-5/6, 5/6, 1/3, -2/3], 24/5-12/5*2^(1/3)], [[-5/6,
5/6, 1/3, -1/2], -1/10*(14-21*2^(1/3))*2^(2/3)], [[-5/6, 5/6, 1/3, 1/2], -1/10*(12-3*2^(1/3))*2^(2/3)], [[-5/6, 5/6, 1/3, 1/3], 6/5-18/5*2^(1/3)], [[-5/6, 5/6, 2/3,
-1/3], 1/10*(3+6*2^(1/3))*2^(2/3)], [[-5/6, 5/6, 2/3, -1/6], 9/10+9/10*2^(1/3)], [[-5/6, 5/6, 2/3, 2/3], -1/20*(-7+6*2^(1/3))*2^(2/3)], [[-5/6, 5/6, 2/3, 5/6], -9/
10+3/5*2^(1/3)], [[-2/3, -5/6, -1/6, -5/6], -1/4*(-729+576*2^(1/3))*2^(2/3)], [[-2/3, -5/6, -1/6, -1/6], 78-243/4*2^(1/3)], [[-2/3, -5/6, -1/6, 1/6], -1/16*(-243+
180*2^(1/3))*2^(2/3)], [[-2/3, -5/6, -1/6, 5/6], -3/4+81/32*2^(1/3)], [[-2/3, -5/6, 5/6, -5/6], -1/112*(243-252*2^(1/3))*2^(2/3)], [[-2/3, -5/6, 5/6, -1/6], 69/28-\
243/224*2^(1/3)], [[-2/3, -5/6, 5/6, 1/6], 81/112*2^(2/3)], [[-2/3, -5/6, 5/6, 5/6], 6/7+81/112*2^(1/3)], [[-2/3, -2/3, -1/2, -1/3], -385/2+154*2^(1/3)], [[-2/3, -2
/3, -1/2, 2/3], -7/2+14/3*2^(1/3)], [[-2/3, -2/3, -1/6, -5/6], -1218/5+972/5*2^(1/3)], [[-2/3, -2/3, -1/6, -1/3], 207/2-81*2^(1/3)], [[-2/3, -2/3, -1/6, 1/6], -21+
18*2^(1/3)], [[-2/3, -2/3, -1/6, 2/3], -3/2+3*2^(1/3)], [[-2/3, -2/3, 1/2, -1/3], 133/22-42/11*2^(1/3)], [[-2/3, -2/3, 1/2, 2/3], 7/22+14/11*2^(1/3)], [[-2/3, -2/3,
1/6, -1/6], 21-140/9*2^(1/3)], [[-2/3, -2/3, 1/6, 5/6], 56/27*2^(1/3)], [[-2/3, -2/3, 5/6, -5/6], 21/5-162/65*2^(1/3)], [[-2/3, -2/3, 5/6, -1/3], 51/26-9/13*2^(1/3)
], [[-2/3, -2/3, 5/6, 1/6], 12/13*2^(1/3)], [[-2/3, -2/3, 5/6, 2/3], 21/26+7/13*2^(1/3)], [[-2/3, -1/2, 1/6, -1/6], -1/8*(91-78*2^(1/3))*2^(2/3)], [[-2/3, -1/2, 1/6
, 5/6], 91/48*2^(2/3)], [[-2/3, -1/3, -1/6, -1/3], -1/10*(-486+378*2^(1/3))*2^(2/3)], [[-2/3, -1/3, -1/6, -1/6], -264/5+216/5*2^(1/3)], [[-2/3, -1/3, -1/6, 2/3], -1
/10*(27-36*2^(1/3))*2^(2/3)], [[-2/3, -1/3, -1/6, 5/6], 42/5-18/5*2^(1/3)], [[-2/3, -1/3, 1/6, -1/2], 312/7-240/7*2^(1/3)], [[-2/3, -1/3, 1/6, -1/3], -1/2*(36-30*2^
(1/3))*2^(2/3)], [[-2/3, -1/3, 1/6, 1/2], -30/7+36/7*2^(1/3)], [[-2/3, -1/3, 1/6, 2/3], -1/14*(-54+24*2^(1/3))*2^(2/3)], [[-2/3, -1/3, 5/6, -1/3], 1/110*(54+18*2^(1
/3))*2^(2/3)], [[-2/3, -1/3, 5/6, -1/6], -6/55+54/55*2^(1/3)], [[-2/3, -1/3, 5/6, 2/3], -1/110*(63-144*2^(1/3))*2^(2/3)], [[-2/3, -1/3, 5/6, 5/6], 228/55-72/55*2^(1
/3)], [[-2/3, -1/6, -1/2, -5/6], -385/2+154*2^(1/3)], [[-2/3, -1/6, -1/2, 1/6], -44+110/3*2^(1/3)], [[-2/3, -1/6, 1/2, -5/6], 22-33/2*2^(1/3)], [[-2/3, -1/6, 1/2, 1
/6], -11/4+55/16*2^(1/3)], [[-2/3, 1/2, 1/6, -1/6], -1/4*(21-21*2^(1/3))*2^(2/3)], [[-2/3, 1/2, 1/6, 5/6], -7/4*2^(2/3)], [[-2/3, 1/3, -1/2, -1/3], -44+110/3*2^(1/3
)], [[-2/3, 1/3, -1/6, -5/6], -264/5+216/5*2^(1/3)], [[-2/3, 1/3, -1/6, -1/3], 36-27*2^(1/3)], [[-2/3, 1/3, -1/6, 1/6], -12+12*2^(1/3)], [[-2/3, 1/3, 1/2, -1/3], 4-\
2*2^(1/3)], [[-2/3, 1/3, 1/6, -1/6], 12-8*2^(1/3)], [[-2/3, 1/3, 1/6, 5/6], -16/3*2^(1/3)], [[-2/3, 1/3, 5/6, -5/6], 12/5-36/35*2^(1/3)], [[-2/3, 1/3, 5/6, -1/3], 
12/7-3/7*2^(1/3)], [[-2/3, 1/3, 5/6, 1/6], 8/7*2^(1/3)], [[-2/3, 1/6, -1/6, -5/6], -1/10*(-486+378*2^(1/3))*2^(2/3)], [[-2/3, 1/6, -1/6, -1/6], 36-27*2^(1/3)], [[-2
/3, 1/6, -1/6, 1/6], -1/8*(-81+54*2^(1/3))*2^(2/3)], [[-2/3, 1/6, 5/6, -5/6], -1/80*(81-108*2^(1/3))*2^(2/3)], [[-2/3, 1/6, 5/6, -1/6], 9/4-27/32*2^(1/3)], [[-2/3,
1/6, 5/6, 1/6], 27/32*2^(2/3)], [[-2/3, 2/3, -1/6, -1/3], -1/8*(-81+54*2^(1/3))*2^(2/3)], [[-2/3, 2/3, -1/6, -1/6], -12+12*2^(1/3)], [[-2/3, 2/3, -1/6, 2/3], -1/8*(
-9+18*2^(1/3))*2^(2/3)], [[-2/3, 2/3, -1/6, 5/6], -3+2^(1/3)], [[-2/3, 2/3, 1/6, -1/2], 12-8*2^(1/3)], [[-2/3, 2/3, 1/6, -1/3], -1/4*(21-21*2^(1/3))*2^(2/3)], [[-2/
3, 2/3, 1/6, 1/2], 3-6*2^(1/3)], [[-2/3, 2/3, 1/6, 2/3], -1/4*(9-3*2^(1/3))*2^(2/3)], [[-2/3, 2/3, 5/6, -1/3], 1/40*(9+18*2^(1/3))*2^(2/3)], [[-2/3, 2/3, 5/6, -1/6]
, 3/5+3/5*2^(1/3)], [[-2/3, 2/3, 5/6, 2/3], -1/40*(-21+18*2^(1/3))*2^(2/3)], [[-2/3, 2/3, 5/6, 5/6], -6/5+4/5*2^(1/3)], [[-2/3, 5/6, -1/2, -5/6], -7/2+14/3*2^(1/3)]
, [[-2/3, 5/6, 1/2, -5/6], 4-2*2^(1/3)], [[-1/2, -5/6, 1/3, -1/6], 77/5-56/5*2^(1/3)], [[-1/2, -5/6, 1/3, 5/6], 7/25+112/75*2^(1/3)], [[-1/2, -2/3, -1/3, -1/3], -1/
2*(-182+143*2^(1/3))*2^(2/3)], [[-1/2, -2/3, -1/3, 2/3], -1/30*(-91+39*2^(1/3))*2^(2/3)], [[-1/2, -2/3, 2/3, -1/3], -1/110*(182-208*2^(1/3))*2^(2/3)], [[-1/2, -2/3,
2/3, 2/3], 1/132*(91+39*2^(1/3))*2^(2/3)], [[-1/2, -1/3, 1/3, -2/3], 286/7-220/7*2^(1/3)], [[-1/2, -1/3, 1/3, 1/3], -11/7+55/21*2^(1/3)], [[-1/2, -1/6, -1/3, -5/6],
-1/2*(-182+143*2^(1/3))*2^(2/3)], [[-1/2, -1/6, -1/3, 1/6], -1/6*(-140+105*2^(1/3))*2^(2/3)], [[-1/2, -1/6, 2/3, -5/6], -1/8*(56-49*2^(1/3))*2^(2/3)], [[-1/2, -1/6,
2/3, 1/6], -1/24*(-40+15*2^(1/3))*2^(2/3)], [[-1/2, 1/3, -1/3, -1/3], -1/6*(-140+105*2^(1/3))*2^(2/3)], [[-1/2, 1/3, 2/3, -1/3], -1/30*(28-42*2^(1/3))*2^(2/3)], [[-\
1/2, 1/6, 1/3, -1/6], 10-20/3*2^(1/3)], [[-1/2, 2/3, 1/3, -2/3], 10-20/3*2^(1/3)], [[-1/2, 5/6, -1/3, -5/6], -1/30*(-91+39*2^(1/3))*2^(2/3)], [[-1/2, 5/6, 2/3, -5/6
], -1/30*(28-42*2^(1/3))*2^(2/3)], [[-1/3, -5/6, 1/2, -1/6], -1/50*(273-247*2^(1/3))*2^(2/3)], [[-1/3, -5/6, 1/2, 5/6], 1/100*(91+26*2^(1/3))*2^(2/3)], [[-1/3, -2/3
, -1/6, -2/3], -165+132*2^(1/3)], [[-1/3, -2/3, -1/6, -1/2], -1/2*(-168+132*2^(1/3))*2^(2/3)], [[-1/3, -2/3, -1/6, 1/2], -1/10*(-72+48*2^(1/3))*2^(2/3)], [[-1/3, -2
/3, -1/6, 1/3], -21+18*2^(1/3)], [[-1/3, -2/3, 1/6, -5/6], -1/14*(972-780*2^(1/3))*2^(2/3)], [[-1/3, -2/3, 1/6, -2/3], 621/7-486/7*2^(1/3)], [[-1/3, -2/3, 1/6, 1/3]
, 45/7-27/7*2^(1/3)], [[-1/3, -2/3, 1/6, 1/6], -1/14*(108-96*2^(1/3))*2^(2/3)], [[-1/3, -2/3, 5/6, -2/3], 57/11-36/11*2^(1/3)], [[-1/3, -2/3, 5/6, -1/2], -1/110*(
168-192*2^(1/3))*2^(2/3)], [[-1/3, -2/3, 5/6, 1/2], 1/110*(72+12*2^(1/3))*2^(2/3)], [[-1/3, -2/3, 5/6, 1/3], -3/55+54/55*2^(1/3)], [[-1/3, -1/2, -1/6, -5/6], -165+
132*2^(1/3)], [[-1/3, -1/2, -1/6, 1/6], -33+55/2*2^(1/3)], [[-1/3, -1/2, 5/6, -5/6], 33/5-22/5*2^(1/3)], [[-1/3, -1/2, 5/6, 1/6], 11/12*2^(1/3)], [[-1/3, -1/3, -1/6
, -5/6], -1/2*(-168+132*2^(1/3))*2^(2/3)], [[-1/3, -1/3, -1/6, 1/6], -1/2*(-40+30*2^(1/3))*2^(2/3)], [[-1/3, -1/3, 1/2, -2/3], -1/2*(30-25*2^(1/3))*2^(2/3)], [[-1/3
, -1/3, 1/2, 1/3], -1/14*(-20+5*2^(1/3))*2^(2/3)], [[-1/3, -1/3, 1/6, -2/3], -1/2*(-81+63*2^(1/3))*2^(2/3)], [[-1/3, -1/3, 1/6, -1/6], -1/2*(36-30*2^(1/3))*2^(2/3)]
, [[-1/3, -1/3, 1/6, 1/3], -1/2*(-6+3*2^(1/3))*2^(2/3)], [[-1/3, -1/3, 1/6, 5/6], 2*2^(2/3)], [[-1/3, -1/3, 5/6, -5/6], -1/18*(56-54*2^(1/3))*2^(2/3)], [[-1/3, -1/3
, 5/6, 1/6], 20/27*2^(2/3)], [[-1/3, -1/6, 1/6, -5/6], -1/2*(-81+63*2^(1/3))*2^(2/3)], [[-1/3, -1/6, 1/6, -1/6], 27-81/4*2^(1/3)], [[-1/3, -1/6, 1/6, 1/6], -1/4*(-\
27+18*2^(1/3))*2^(2/3)], [[-1/3, -1/6, 1/6, 5/6], 27/8*2^(1/3)], [[-1/3, 1/2, -1/6, -5/6], -21+18*2^(1/3)], [[-1/3, 1/2, -1/6, 1/6], -15+15*2^(1/3)], [[-1/3, 1/2, 5
/6, -5/6], 3-3/2*2^(1/3)], [[-1/3, 1/2, 5/6, 1/6], 5/4*2^(1/3)], [[-1/3, 1/3, -1/6, -2/3], -33+55/2*2^(1/3)], [[-1/3, 1/3, -1/6, -1/2], -1/2*(-40+30*2^(1/3))*2^(2/3
)], [[-1/3, 1/3, -1/6, 1/2], -1/2*(-24+12*2^(1/3))*2^(2/3)], [[-1/3, 1/3, -1/6, 1/3], -15+15*2^(1/3)], [[-1/3, 1/3, 1/6, -5/6], -1/2*(36-30*2^(1/3))*2^(2/3)], [[-1/
3, 1/3, 1/6, -2/3], 27-81/4*2^(1/3)], [[-1/3, 1/3, 1/6, 1/3], 9-9/2*2^(1/3)], [[-1/3, 1/3, 1/6, 1/6], -1/2*(12-12*2^(1/3))*2^(2/3)], [[-1/3, 1/3, 5/6, -2/3], 3-3/2*
2^(1/3)], [[-1/3, 1/3, 5/6, -1/2], -1/10*(8-12*2^(1/3))*2^(2/3)], [[-1/3, 1/3, 5/6, 1/2], -1/10*(-24+6*2^(1/3))*2^(2/3)], [[-1/3, 1/3, 5/6, 1/3], -3/5+9/5*2^(1/3)],
[[-1/3, 1/6, 1/2, -1/6], -1/2*(7-7*2^(1/3))*2^(2/3)], [[-1/3, 2/3, -1/6, -5/6], -1/10*(-72+48*2^(1/3))*2^(2/3)], [[-1/3, 2/3, -1/6, 1/6], -1/2*(-24+12*2^(1/3))*2^(2
/3)], [[-1/3, 2/3, 1/2, -2/3], -1/2*(7-7*2^(1/3))*2^(2/3)], [[-1/3, 2/3, 1/6, -2/3], -1/4*(-27+18*2^(1/3))*2^(2/3)], [[-1/3, 2/3, 1/6, -1/6], -1/2*(12-12*2^(1/3))*2
^(2/3)], [[-1/3, 2/3, 1/6, 5/6], -2/3*2^(2/3)], [[-1/3, 2/3, 5/6, -5/6], -1/10*(8-12*2^(1/3))*2^(2/3)], [[-1/3, 2/3, 5/6, 1/6], 4/3*2^(2/3)], [[-1/3, 5/6, 1/6, -5/6
], -1/2*(-6+3*2^(1/3))*2^(2/3)], [[-1/3, 5/6, 1/6, -1/6], 9-9/2*2^(1/3)], [[-1/3, 5/6, 1/6, 5/6], -3/8*2^(1/3)], [[-1/6, -5/6, 1/3, -5/6], -1/20*(972-783*2^(1/3))*2
^(2/3)], [[-1/6, -5/6, 1/3, -2/3], 312/5-243/5*2^(1/3)], [[-1/6, -5/6, 1/3, 1/3], 24/5-27/10*2^(1/3)], [[-1/6, -5/6, 1/3, 1/6], -1/20*(108-99*2^(1/3))*2^(2/3)], [[-\
1/6, -5/6, 2/3, -1/2], 66/5-48/5*2^(1/3)], [[-1/6, -5/6, 2/3, -1/3], -1/50*(252-228*2^(1/3))*2^(2/3)], [[-1/6, -5/6, 2/3, 1/2], -12/25+36/25*2^(1/3)], [[-1/6, -5/6,
2/3, 2/3], -1/50*(-54+6*2^(1/3))*2^(2/3)], [[-1/6, -2/3, 2/3, -2/3], 33/2-110/9*2^(1/3)], [[-1/6, -2/3, 2/3, 1/3], 55/54*2^(1/3)], [[-1/6, -1/2, 2/3, -2/3], -1/4*(
35-30*2^(1/3))*2^(2/3)], [[-1/6, -1/2, 2/3, 1/3], 5/6*2^(2/3)], [[-1/6, -1/3, 1/3, -2/3], -198/5+162/5*2^(1/3)], [[-1/6, -1/3, 1/3, -1/3], -1/10*(162-135*2^(1/3))*2
^(2/3)], [[-1/6, -1/3, 1/3, 1/3], -9/5+27/10*2^(1/3)], [[-1/6, -1/3, 1/3, 2/3], 27/20*2^(2/3)], [[-1/6, -1/6, 1/3, -5/6], -198/5+162/5*2^(1/3)], [[-1/6, -1/6, 1/3,
-1/3], 24-18*2^(1/3)], [[-1/6, -1/6, 1/3, 1/6], -6+6*2^(1/3)], [[-1/6, -1/6, 1/3, 2/3], 2*2^(1/3)], [[-1/6, 1/2, 2/3, -2/3], -1/2*(6-6*2^(1/3))*2^(2/3)], [[-1/6, 1/
2, 2/3, 1/3], 2*2^(2/3)], [[-1/6, 1/3, 2/3, -2/3], 15/2-5*2^(1/3)], [[-1/6, 1/3, 2/3, 1/3], 5/3*2^(1/3)], [[-1/6, 1/6, 1/3, -5/6], -1/10*(162-135*2^(1/3))*2^(2/3)],
[[-1/6, 1/6, 1/3, -2/3], 24-18*2^(1/3)], [[-1/6, 1/6, 1/3, 1/3], 6-3*2^(1/3)], [[-1/6, 1/6, 1/3, 1/6], -1/2*(9-9*2^(1/3))*2^(2/3)], [[-1/6, 1/6, 2/3, -1/2], 15/2-5*
2^(1/3)], [[-1/6, 1/6, 2/3, -1/3], -1/2*(6-6*2^(1/3))*2^(2/3)], [[-1/6, 1/6, 2/3, 1/2], -3/2+3*2^(1/3)], [[-1/6, 1/6, 2/3, 2/3], -1/2*(-9+3*2^(1/3))*2^(2/3)], [[-1/
6, 2/3, 1/3, -2/3], -6+6*2^(1/3)], [[-1/6, 2/3, 1/3, -1/3], -1/2*(9-9*2^(1/3))*2^(2/3)], [[-1/6, 2/3, 1/3, 2/3], -3/4*2^(2/3)], [[-1/6, 5/6, 1/3, -5/6], -9/5+27/10*
2^(1/3)], [[-1/6, 5/6, 1/3, -1/3], 6-3*2^(1/3)], [[-1/6, 5/6, 1/3, 2/3], -1/3*2^(1/3)], [[1/2, -2/3, 2/3, -1/3], -1/10*(-42+28*2^(1/3))*2^(2/3)], [[1/2, -2/3, 2/3,
2/3], 1/20*(7+7*2^(1/3))*2^(2/3)], [[1/2, -1/6, 2/3, -5/6], -1/10*(-42+28*2^(1/3))*2^(2/3)], [[1/2, -1/6, 2/3, 1/6], -1/2*(-4+2*2^(1/3))*2^(2/3)], [[1/2, 1/3, 2/3,
-1/3], -1/2*(-4+2*2^(1/3))*2^(2/3)], [[1/2, 5/6, 2/3, -5/6], 1/20*(7+7*2^(1/3))*2^(2/3)], [[1/3, -5/6, 5/6, -5/6], -1/20*(-81+54*2^(1/3))*2^(2/3)], [[1/3, -5/6, 5/6
, -1/6], 18/5-81/40*2^(1/3)], [[1/3, -5/6, 5/6, 1/6], 27/40*2^(2/3)], [[1/3, -5/6, 5/6, 5/6], 9/10+27/80*2^(1/3)], [[1/3, -2/3, 1/2, -1/3], -14+12*2^(1/3)], [[1/3,
-2/3, 1/2, 2/3], 2/5+4/5*2^(1/3)], [[1/3, -2/3, 5/6, -5/6], -24/5+162/35*2^(1/3)], [[1/3, -2/3, 5/6, -1/3], 30/7-18/7*2^(1/3)], [[1/3, -2/3, 5/6, 1/6], 6/7*2^(1/3)]
, [[1/3, -2/3, 5/6, 2/3], 6/7+2/7*2^(1/3)], [[1/3, -1/3, 5/6, -1/3], -1/10*(-18+9*2^(1/3))*2^(2/3)], [[1/3, -1/3, 5/6, -1/6], -6/5+9/5*2^(1/3)], [[1/3, -1/3, 5/6, 2
/3], -1/10*(3-9*2^(1/3))*2^(2/3)], [[1/3, -1/3, 5/6, 5/6], 12/5-3/5*2^(1/3)], [[1/3, -1/6, 1/2, -5/6], -14+12*2^(1/3)], [[1/3, -1/6, 1/2, 1/6], -5+5*2^(1/3)], [[1/3
, 1/3, 1/2, -1/3], -5+5*2^(1/3)], [[1/3, 1/3, 5/6, -5/6], -6/5+9/5*2^(1/3)], [[1/3, 1/3, 5/6, -1/3], 3-3/2*2^(1/3)], [[1/3, 1/3, 5/6, 1/6], 2^(1/3)], [[1/3, 1/6, 5/
6, -5/6], -1/10*(-18+9*2^(1/3))*2^(2/3)], [[1/3, 1/6, 5/6, -1/6], 3-3/2*2^(1/3)], [[1/3, 1/6, 5/6, 1/6], 3/4*2^(2/3)], [[1/3, 2/3, 5/6, -1/3], 3/4*2^(2/3)], [[1/3,
2/3, 5/6, -1/6], 2^(1/3)], [[1/3, 2/3, 5/6, 2/3], 1/4*2^(2/3)], [[1/3, 2/3, 5/6, 5/6], 1/3*2^(1/3)], [[1/3, 5/6, 1/2, -5/6], 2/5+4/5*2^(1/3)], [[1/6, -5/6, 2/3, -2/
3], -1/8*(-81+60*2^(1/3))*2^(2/3)], [[1/6, -5/6, 2/3, -1/6], -1/8*(36-33*2^(1/3))*2^(2/3)], [[1/6, -5/6, 2/3, 1/3], 3/4*2^(2/3)], [[1/6, -5/6, 2/3, 5/6], 1/8*(4+3*2
^(1/3))*2^(2/3)], [[1/6, -2/3, 2/3, -2/3], -27/2+81/7*2^(1/3)], [[1/6, -2/3, 2/3, -1/3], -1/14*(81-72*2^(1/3))*2^(2/3)], [[1/6, -2/3, 2/3, 1/3], 27/28*2^(1/3)], [[1
/6, -2/3, 2/3, 2/3], 1/56*(27+18*2^(1/3))*2^(2/3)], [[1/6, -1/2, 1/3, -1/3], -24+20*2^(1/3)], [[1/6, -1/2, 1/3, 2/3], 4/3*2^(1/3)], [[1/6, -1/3, 1/3, -1/3], -1/2*(-\
28+21*2^(1/3))*2^(2/3)], [[1/6, -1/3, 1/3, 2/3], 7/6*2^(2/3)], [[1/6, -1/6, 1/3, -2/3], -24+20*2^(1/3)], [[1/6, -1/6, 1/3, -1/2], -1/2*(-28+21*2^(1/3))*2^(2/3)], [[
1/6, -1/6, 1/3, 1/2], -1/2*(-6+3*2^(1/3))*2^(2/3)], [[1/6, -1/6, 1/3, 1/3], -6+6*2^(1/3)], [[1/6, -1/6, 2/3, -1/3], -1/2*(-9+6*2^(1/3))*2^(2/3)], [[1/6, -1/6, 2/3,
-1/6], -9/2+9/2*2^(1/3)], [[1/6, -1/6, 2/3, 2/3], -1/4*(3-6*2^(1/3))*2^(2/3)], [[1/6, -1/6, 2/3, 5/6], 9/2-3/2*2^(1/3)], [[1/6, 1/2, 1/3, -1/3], -6+6*2^(1/3)], [[1/
6, 1/2, 1/3, 2/3], -2*2^(1/3)], [[1/6, 1/3, 2/3, -2/3], -9/2+9/2*2^(1/3)], [[1/6, 1/3, 2/3, -1/3], -1/2*(6-6*2^(1/3))*2^(2/3)], [[1/6, 1/3, 2/3, 1/3], 3/2*2^(1/3)],
[[1/6, 1/6, 2/3, -2/3], -1/2*(-9+6*2^(1/3))*2^(2/3)], [[1/6, 1/6, 2/3, -1/6], -1/2*(6-6*2^(1/3))*2^(2/3)], [[1/6, 1/6, 2/3, 1/3], 2^(2/3)], [[1/6, 2/3, 1/3, -1/3],
-1/2*(-6+3*2^(1/3))*2^(2/3)], [[1/6, 2/3, 1/3, 2/3], -1/2*2^(2/3)], [[1/6, 5/6, 1/3, -2/3], 4/3*2^(1/3)], [[1/6, 5/6, 1/3, -1/2], 7/6*2^(2/3)], [[1/6, 5/6, 1/3, 1/2
], -1/2*2^(2/3)], [[1/6, 5/6, 1/3, 1/3], -2*2^(1/3)], [[1/6, 5/6, 2/3, -1/3], 2^(2/3)], [[1/6, 5/6, 2/3, -1/6], 3/2*2^(1/3)], [[1/6, 5/6, 2/3, 2/3], 1/6*2^(2/3)], [
[1/6, 5/6, 2/3, 5/6], 1/4*2^(1/3)], [[2/3, -2/3, 5/6, -2/3], -3/2+2*2^(1/3)], [[2/3, -2/3, 5/6, -1/2], -1/10*(-14+6*2^(1/3))*2^(2/3)], [[2/3, -2/3, 5/6, 1/2], 1/10*
(3+3*2^(1/3))*2^(2/3)], [[2/3, -2/3, 5/6, 1/3], 3/10+3/5*2^(1/3)], [[2/3, -1/2, 5/6, -5/6], -3/2+2*2^(1/3)], [[2/3, -1/2, 5/6, 1/6], 5/6*2^(1/3)], [[2/3, -1/3, 5/6,
-5/6], -1/10*(-14+6*2^(1/3))*2^(2/3)], [[2/3, -1/3, 5/6, 1/6], 2/3*2^(2/3)], [[2/3, 1/2, 5/6, -5/6], 3/10+3/5*2^(1/3)], [[2/3, 1/2, 5/6, 1/6], 2^(1/3)], [[2/3, 1/3,
5/6, -2/3], 5/6*2^(1/3)], [[2/3, 1/3, 5/6, -1/2], 2/3*2^(2/3)], [[2/3, 1/3, 5/6, 1/2], 2^(2/3)], [[2/3, 1/3, 5/6, 1/3], 2^(1/3)], [[2/3, 2/3, 5/6, -5/6], 1/10*(3+3*
2^(1/3))*2^(2/3)], [[2/3, 2/3, 5/6, 1/6], 2^(2/3)]]]:
end:

#end pre-computed

#OPEZ2(a1,a2,b1,b2,n,N): The second-order linear recurrence equation annihilated by the Generalized
#Beukers integral
#print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
#Here N is the shift operator in n, Nf(n):=f(n+1).
#Try:
#For the original Apery Recurrence for Zeta(2) type: OPEZ2(0,0,0,0,n,N);
#For the Zudilin (2003) recurrence for Catalan's constant type: OPEZ2(1/2,0,0,1/2,n,N);

OPEZ2:=proc(a1,a2,b1,b2,n,N) local ope,i:
ope:=-(b2-n-1)*(a2-n-1)*(a2+b2-n-1)*(b1-n-1)*(a1-1-n)*(a1*a2*b1+a1*a2*b2-2*a1*a2*n+
a1*b1*b2-2*a1*b1*n+a1*b2^2-3*a1*b2*n+3*a1*n^2+a2^2*b1+a2^2*b2-2*a2^2*n+a2*b1*b2
-3*a2*b1*n+a2*b2^2-5*a2*b2*n+6*a2*n^2-2*b1*b2*n+3*b1*n^2-2*b2^2*n+6*b2*n^2-5*n^
3-4*a1*a2-4*a1*b1-6*a1*b2+12*a1*n-4*a2^2-6*a2*b1-10*a2*b2+24*a2*n-4*b1*b2+12*b1
*n-4*b2^2+24*b2*n-30*n^2+12*a1+24*a2+12*b1+24*b2-60*n-40)/(2+n)/(b1+b2-n-2)/(n+
2+a1-b1-b2)/(a1+a2-n-2)/(-n-2+a1+a2-b1)/(a1*a2*b1+a1*a2*b2-2*a1*a2*n+a1*b1*b2-2
*a1*b1*n+a1*b2^2-3*a1*b2*n+3*a1*n^2+a2^2*b1+a2^2*b2-2*a2^2*n+a2*b1*b2-3*a2*b1*n
+a2*b2^2-5*a2*b2*n+6*a2*n^2-2*b1*b2*n+3*b1*n^2-2*b2^2*n+6*b2*n^2-5*n^3-2*a1*a2-\
2*a1*b1-3*a1*b2+6*a1*n-2*a2^2-3*a2*b1-5*a2*b2+12*a2*n-2*b1*b2+6*b1*n-2*b2^2+12*
b2*n-15*n^2+3*a1+6*a2+3*b1+6*b2-15*n-5)+(a1^3*a2^2*b1^2*b2-2*a1^3*a2^2*b1^2*n+2
*a1^3*a2^2*b1*b2^2-6*a1^3*a2^2*b1*b2*n+6*a1^3*a2^2*b1*n^2+a1^3*a2^2*b2^3-4*a1^3
*a2^2*b2^2*n+6*a1^3*a2^2*b2*n^2-4*a1^3*a2^2*n^3+2*a1^3*a2*b1^2*b2^2-6*a1^3*a2*
b1^2*b2*n+6*a1^3*a2*b1^2*n^2+4*a1^3*a2*b1*b2^3-16*a1^3*a2*b1*b2^2*n+24*a1^3*a2*
b1*b2*n^2-16*a1^3*a2*b1*n^3+2*a1^3*a2*b2^4-10*a1^3*a2*b2^3*n+20*a1^3*a2*b2^2*n^
2-20*a1^3*a2*b2*n^3+10*a1^3*a2*n^4+a1^3*b1^2*b2^3-4*a1^3*b1^2*b2^2*n+6*a1^3*b1^
2*b2*n^2-4*a1^3*b1^2*n^3+2*a1^3*b1*b2^4-10*a1^3*b1*b2^3*n+20*a1^3*b1*b2^2*n^2-\
20*a1^3*b1*b2*n^3+10*a1^3*b1*n^4+a1^3*b2^5-6*a1^3*b2^4*n+15*a1^3*b2^3*n^2-20*a1
^3*b2^2*n^3+15*a1^3*b2*n^4-6*a1^3*n^5+a1^2*a2^3*b1^3+5*a1^2*a2^3*b1^2*b2-10*a1^
2*a2^3*b1^2*n+7*a1^2*a2^3*b1*b2^2-24*a1^2*a2^3*b1*b2*n+24*a1^2*a2^3*b1*n^2+3*a1
^2*a2^3*b2^3-14*a1^2*a2^3*b2^2*n+24*a1^2*a2^3*b2*n^2-16*a1^2*a2^3*n^3+2*a1^2*a2
^2*b1^3*b2-4*a1^2*a2^2*b1^3*n+10*a1^2*a2^2*b1^2*b2^2-40*a1^2*a2^2*b1^2*b2*n+43*
a1^2*a2^2*b1^2*n^2+14*a1^2*a2^2*b1*b2^3-76*a1^2*a2^2*b1*b2^2*n+144*a1^2*a2^2*b1
*b2*n^2-102*a1^2*a2^2*b1*n^3+6*a1^2*a2^2*b2^4-40*a1^2*a2^2*b2^3*n+103*a1^2*a2^2
*b2^2*n^2-126*a1^2*a2^2*b2*n^3+66*a1^2*a2^2*n^4+a1^2*a2*b1^3*b2^2-6*a1^2*a2*b1^
3*b2*n+6*a1^2*a2*b1^3*n^2+5*a1^2*a2*b1^2*b2^3-40*a1^2*a2*b1^2*b2^2*n+90*a1^2*a2
*b1^2*b2*n^2-66*a1^2*a2*b1^2*n^3+7*a1^2*a2*b1*b2^4-66*a1^2*a2*b1*b2^3*n+203*a1^
2*a2*b1*b2^2*n^2-274*a1^2*a2*b1*b2*n^3+149*a1^2*a2*b1*n^4+3*a1^2*a2*b2^5-32*a1^
2*a2*b2^4*n+121*a1^2*a2*b2^3*n^2-224*a1^2*a2*b2^2*n^3+215*a1^2*a2*b2*n^4-92*a1^
2*a2*n^5-2*a1^2*b1^3*b2^2*n+6*a1^2*b1^3*b2*n^2-4*a1^2*b1^3*n^3-10*a1^2*b1^2*b2^
3*n+43*a1^2*b1^2*b2^2*n^2-66*a1^2*b1^2*b2*n^3+36*a1^2*b1^2*n^4-14*a1^2*b1*b2^4*
n+76*a1^2*b1*b2^3*n^2-164*a1^2*b1*b2^2*n^3+170*a1^2*b1*b2*n^4-74*a1^2*b1*n^5-6*
a1^2*b2^5*n+39*a1^2*b2^4*n^2-106*a1^2*b2^3*n^3+153*a1^2*b2^2*n^4-120*a1^2*b2*n^
5+43*a1^2*n^6+2*a1*a2^4*b1^3+7*a1*a2^4*b1^2*b2-14*a1*a2^4*b1^2*n+8*a1*a2^4*b1*
b2^2-30*a1*a2^4*b1*b2*n+30*a1*a2^4*b1*n^2+3*a1*a2^4*b2^3-16*a1*a2^4*b2^2*n+30*
a1*a2^4*b2*n^2-20*a1*a2^4*n^3+4*a1*a2^3*b1^3*b2-10*a1*a2^3*b1^3*n+14*a1*a2^3*b1
^2*b2^2-66*a1*a2^3*b1^2*b2*n+76*a1*a2^3*b1^2*n^2+16*a1*a2^3*b1*b2^3-106*a1*a2^3
*b1*b2^2*n+228*a1*a2^3*b1*b2*n^2-168*a1*a2^3*b1*n^3+6*a1*a2^3*b2^4-50*a1*a2^3*
b2^3*n+152*a1*a2^3*b2^2*n^2-208*a1*a2^3*b2*n^3+112*a1*a2^3*n^4+2*a1*a2^2*b1^3*
b2^2-16*a1*a2^2*b1^3*b2*n+20*a1*a2^2*b1^3*n^2+7*a1*a2^2*b1^2*b2^3-76*a1*a2^2*b1
^2*b2^2*n+203*a1*a2^2*b1^2*b2*n^2-164*a1*a2^2*b1^2*n^3+8*a1*a2^2*b1*b2^4-106*a1
*a2^2*b1*b2^3*n+404*a1*a2^2*b1*b2^2*n^2-630*a1*a2^2*b1*b2*n^3+366*a1*a2^2*b1*n^
4+3*a1*a2^2*b2^5-46*a1*a2^2*b2^4*n+221*a1*a2^2*b2^3*n^2-490*a1*a2^2*b2^2*n^3+
534*a1*a2^2*b2*n^4-240*a1*a2^2*n^5-6*a1*a2*b1^3*b2^2*n+24*a1*a2*b1^3*b2*n^2-20*
a1*a2*b1^3*n^3-24*a1*a2*b1^2*b2^3*n+144*a1*a2*b1^2*b2^2*n^2-274*a1*a2*b1^2*b2*n
^3+170*a1*a2*b1^2*n^4-30*a1*a2*b1*b2^4*n+228*a1*a2*b1*b2^3*n^2-630*a1*a2*b1*b2^
2*n^3+776*a1*a2*b1*b2*n^4-370*a1*a2*b1*n^5-12*a1*a2*b2^5*n+108*a1*a2*b2^4*n^2-\
380*a1*a2*b2^3*n^3+672*a1*a2*b2^2*n^4-610*a1*a2*b2*n^5+234*a1*a2*n^6+6*a1*b1^3*
b2^2*n^2-16*a1*b1^3*b2*n^3+10*a1*b1^3*n^4+24*a1*b1^2*b2^3*n^2-102*a1*b1^2*b2^2*
n^3+149*a1*b1^2*b2*n^4-74*a1*b1^2*n^5+30*a1*b1*b2^4*n^2-168*a1*b1*b2^3*n^3+366*
a1*b1*b2^2*n^4-370*a1*b1*b2*n^5+148*a1*b1*n^6+12*a1*b2^5*n^2-82*a1*b2^4*n^3+232
*a1*b2^3*n^4-342*a1*b2^2*n^5+265*a1*b2*n^6-88*a1*n^7+a2^5*b1^3+3*a2^5*b1^2*b2-6
*a2^5*b1^2*n+3*a2^5*b1*b2^2-12*a2^5*b1*b2*n+12*a2^5*b1*n^2+a2^5*b2^3-6*a2^5*b2^
2*n+12*a2^5*b2*n^2-8*a2^5*n^3+2*a2^4*b1^3*b2-6*a2^4*b1^3*n+6*a2^4*b1^2*b2^2-32*
a2^4*b1^2*b2*n+39*a2^4*b1^2*n^2+6*a2^4*b1*b2^3-46*a2^4*b1*b2^2*n+108*a2^4*b1*b2
*n^2-82*a2^4*b1*n^3+2*a2^4*b2^4-20*a2^4*b2^3*n+69*a2^4*b2^2*n^2-102*a2^4*b2*n^3
+56*a2^4*n^4+a2^3*b1^3*b2^2-10*a2^3*b1^3*b2*n+15*a2^3*b1^3*n^2+3*a2^3*b1^2*b2^3
-40*a2^3*b1^2*b2^2*n+121*a2^3*b1^2*b2*n^2-106*a2^3*b1^2*n^3+3*a2^3*b1*b2^4-50*
a2^3*b1*b2^3*n+221*a2^3*b1*b2^2*n^2-380*a2^3*b1*b2*n^3+232*a2^3*b1*n^4+a2^3*b2^
5-20*a2^3*b2^4*n+115*a2^3*b2^3*n^2-290*a2^3*b2^2*n^3+344*a2^3*b2*n^4-160*a2^3*n
^5-4*a2^2*b1^3*b2^2*n+20*a2^2*b1^3*b2*n^2-20*a2^2*b1^3*n^3-14*a2^2*b1^2*b2^3*n+
103*a2^2*b1^2*b2^2*n^2-224*a2^2*b1^2*b2*n^3+153*a2^2*b1^2*n^4-16*a2^2*b1*b2^4*n
+152*a2^2*b1*b2^3*n^2-490*a2^2*b1*b2^2*n^3+672*a2^2*b1*b2*n^4-342*a2^2*b1*n^5-6
*a2^2*b2^5*n+69*a2^2*b2^4*n^2-290*a2^2*b2^3*n^3+585*a2^2*b2^2*n^4-582*a2^2*b2*n
^5+234*a2^2*n^6+6*a2*b1^3*b2^2*n^2-20*a2*b1^3*b2*n^3+15*a2*b1^3*n^4+24*a2*b1^2*
b2^3*n^2-126*a2*b1^2*b2^2*n^3+215*a2*b1^2*b2*n^4-120*a2*b1^2*n^5+30*a2*b1*b2^4*
n^2-208*a2*b1*b2^3*n^3+534*a2*b1*b2^2*n^4-610*a2*b1*b2*n^5+265*a2*b1*n^6+12*a2*
b2^5*n^2-102*a2*b2^4*n^3+344*a2*b2^3*n^4-582*a2*b2^2*n^5+499*a2*b2*n^6-176*a2*n
^7-4*b1^3*b2^2*n^3+10*b1^3*b2*n^4-6*b1^3*n^5-16*b1^2*b2^3*n^3+66*b1^2*b2^2*n^4-\
92*b1^2*b2*n^5+43*b1^2*n^6-20*b1*b2^4*n^3+112*b1*b2^3*n^4-240*b1*b2^2*n^5+234*
b1*b2*n^6-88*b1*n^7-8*b2^5*n^3+56*b2^4*n^4-160*b2^3*n^5+234*b2^2*n^6-176*b2*n^7
+55*n^8-3*a1^3*a2^2*b1^2-9*a1^3*a2^2*b1*b2+18*a1^3*a2^2*b1*n-6*a1^3*a2^2*b2^2+
18*a1^3*a2^2*b2*n-18*a1^3*a2^2*n^2-9*a1^3*a2*b1^2*b2+18*a1^3*a2*b1^2*n-24*a1^3*
a2*b1*b2^2+72*a1^3*a2*b1*b2*n-72*a1^3*a2*b1*n^2-15*a1^3*a2*b2^3+60*a1^3*a2*b2^2
*n-90*a1^3*a2*b2*n^2+60*a1^3*a2*n^3-6*a1^3*b1^2*b2^2+18*a1^3*b1^2*b2*n-18*a1^3*
b1^2*n^2-15*a1^3*b1*b2^3+60*a1^3*b1*b2^2*n-90*a1^3*b1*b2*n^2+60*a1^3*b1*n^3-9*
a1^3*b2^4+45*a1^3*b2^3*n-90*a1^3*b2^2*n^2+90*a1^3*b2*n^3-45*a1^3*n^4-15*a1^2*a2
^3*b1^2-36*a1^2*a2^3*b1*b2+72*a1^2*a2^3*b1*n-21*a1^2*a2^3*b2^2+72*a1^2*a2^3*b2*
n-72*a1^2*a2^3*n^2-6*a1^2*a2^2*b1^3-60*a1^2*a2^2*b1^2*b2+129*a1^2*a2^2*b1^2*n-\
114*a1^2*a2^2*b1*b2^2+432*a1^2*a2^2*b1*b2*n-459*a1^2*a2^2*b1*n^2-60*a1^2*a2^2*
b2^3+309*a1^2*a2^2*b2^2*n-567*a1^2*a2^2*b2*n^2+396*a1^2*a2^2*n^3-9*a1^2*a2*b1^3
*b2+18*a1^2*a2*b1^3*n-60*a1^2*a2*b1^2*b2^2+270*a1^2*a2*b1^2*b2*n-297*a1^2*a2*b1
^2*n^2-99*a1^2*a2*b1*b2^3+609*a1^2*a2*b1*b2^2*n-1233*a1^2*a2*b1*b2*n^2+894*a1^2
*a2*b1*n^3-48*a1^2*a2*b2^4+363*a1^2*a2*b2^3*n-1008*a1^2*a2*b2^2*n^2+1290*a1^2*
a2*b2*n^3-690*a1^2*a2*n^4-3*a1^2*b1^3*b2^2+18*a1^2*b1^3*b2*n-18*a1^2*b1^3*n^2-\
15*a1^2*b1^2*b2^3+129*a1^2*b1^2*b2^2*n-297*a1^2*b1^2*b2*n^2+216*a1^2*b1^2*n^3-\
21*a1^2*b1*b2^4+228*a1^2*b1*b2^3*n-738*a1^2*b1*b2^2*n^2+1020*a1^2*b1*b2*n^3-555
*a1^2*b1*n^4-9*a1^2*b2^5+117*a1^2*b2^4*n-477*a1^2*b2^3*n^2+918*a1^2*b2^2*n^3-\
900*a1^2*b2*n^4+387*a1^2*n^5-21*a1*a2^4*b1^2-45*a1*a2^4*b1*b2+90*a1*a2^4*b1*n-\
24*a1*a2^4*b2^2+90*a1*a2^4*b2*n-90*a1*a2^4*n^2-15*a1*a2^3*b1^3-99*a1*a2^3*b1^2*
b2+228*a1*a2^3*b1^2*n-159*a1*a2^3*b1*b2^2+684*a1*a2^3*b1*b2*n-756*a1*a2^3*b1*n^
2-75*a1*a2^3*b2^3+456*a1*a2^3*b2^2*n-936*a1*a2^3*b2*n^2+672*a1*a2^3*n^3-24*a1*
a2^2*b1^3*b2+60*a1*a2^2*b1^3*n-114*a1*a2^2*b1^2*b2^2+609*a1*a2^2*b1^2*b2*n-738*
a1*a2^2*b1^2*n^2-159*a1*a2^2*b1*b2^3+1212*a1*a2^2*b1*b2^2*n-2835*a1*a2^2*b1*b2*
n^2+2196*a1*a2^2*b1*n^3-69*a1*a2^2*b2^4+663*a1*a2^2*b2^3*n-2205*a1*a2^2*b2^2*n^
2+3204*a1*a2^2*b2*n^3-1800*a1*a2^2*n^4-9*a1*a2*b1^3*b2^2+72*a1*a2*b1^3*b2*n-90*
a1*a2*b1^3*n^2-36*a1*a2*b1^2*b2^3+432*a1*a2*b1^2*b2^2*n-1233*a1*a2*b1^2*b2*n^2+
1020*a1*a2*b1^2*n^3-45*a1*a2*b1*b2^4+684*a1*a2*b1*b2^3*n-2835*a1*a2*b1*b2^2*n^2
+4656*a1*a2*b1*b2*n^3-2775*a1*a2*b1*n^4-18*a1*a2*b2^5+324*a1*a2*b2^4*n-1710*a1*
a2*b2^3*n^2+4032*a1*a2*b2^2*n^3-4575*a1*a2*b2*n^4+2106*a1*a2*n^5+18*a1*b1^3*b2^
2*n-72*a1*b1^3*b2*n^2+60*a1*b1^3*n^3+72*a1*b1^2*b2^3*n-459*a1*b1^2*b2^2*n^2+894
*a1*b1^2*b2*n^3-555*a1*b1^2*n^4+90*a1*b1*b2^4*n-756*a1*b1*b2^3*n^2+2196*a1*b1*
b2^2*n^3-2775*a1*b1*b2*n^4+1332*a1*b1*n^5+36*a1*b2^5*n-369*a1*b2^4*n^2+1392*a1*
b2^3*n^3-2565*a1*b2^2*n^4+2385*a1*b2*n^5-924*a1*n^6-9*a2^5*b1^2-18*a2^5*b1*b2+
36*a2^5*b1*n-9*a2^5*b2^2+36*a2^5*b2*n-36*a2^5*n^2-9*a2^4*b1^3-48*a2^4*b1^2*b2+
117*a2^4*b1^2*n-69*a2^4*b1*b2^2+324*a2^4*b1*b2*n-369*a2^4*b1*n^2-30*a2^4*b2^3+
207*a2^4*b2^2*n-459*a2^4*b2*n^2+336*a2^4*n^3-15*a2^3*b1^3*b2+45*a2^3*b1^3*n-60*
a2^3*b1^2*b2^2+363*a2^3*b1^2*b2*n-477*a2^3*b1^2*n^2-75*a2^3*b1*b2^3+663*a2^3*b1
*b2^2*n-1710*a2^3*b1*b2*n^2+1392*a2^3*b1*n^3-30*a2^3*b2^4+345*a2^3*b2^3*n-1305*
a2^3*b2^2*n^2+2064*a2^3*b2*n^3-1200*a2^3*n^4-6*a2^2*b1^3*b2^2+60*a2^2*b1^3*b2*n
-90*a2^2*b1^3*n^2-21*a2^2*b1^2*b2^3+309*a2^2*b1^2*b2^2*n-1008*a2^2*b1^2*b2*n^2+
918*a2^2*b1^2*n^3-24*a2^2*b1*b2^4+456*a2^2*b1*b2^3*n-2205*a2^2*b1*b2^2*n^2+4032
*a2^2*b1*b2*n^3-2565*a2^2*b1*n^4-9*a2^2*b2^5+207*a2^2*b2^4*n-1305*a2^2*b2^3*n^2
+3510*a2^2*b2^2*n^3-4365*a2^2*b2*n^4+2106*a2^2*n^5+18*a2*b1^3*b2^2*n-90*a2*b1^3
*b2*n^2+90*a2*b1^3*n^3+72*a2*b1^2*b2^3*n-567*a2*b1^2*b2^2*n^2+1290*a2*b1^2*b2*n
^3-900*a2*b1^2*n^4+90*a2*b1*b2^4*n-936*a2*b1*b2^3*n^2+3204*a2*b1*b2^2*n^3-4575*
a2*b1*b2*n^4+2385*a2*b1*n^5+36*a2*b2^5*n-459*a2*b2^4*n^2+2064*a2*b2^3*n^3-4365*
a2*b2^2*n^4+4491*a2*b2*n^5-1848*a2*n^6-18*b1^3*b2^2*n^2+60*b1^3*b2*n^3-45*b1^3*
n^4-72*b1^2*b2^3*n^2+396*b1^2*b2^2*n^3-690*b1^2*b2*n^4+387*b1^2*n^5-90*b1*b2^4*
n^2+672*b1*b2^3*n^3-1800*b1*b2^2*n^4+2106*b1*b2*n^5-924*b1*n^6-36*b2^5*n^2+336*
b2^4*n^3-1200*b2^3*n^4+2106*b2^2*n^5-1848*b2*n^6+660*n^7+13*a1^3*a2^2*b1+13*a1^
3*a2^2*b2-26*a1^3*a2^2*n+13*a1^3*a2*b1^2+53*a1^3*a2*b1*b2-106*a1^3*a2*b1*n+44*
a1^3*a2*b2^2-132*a1^3*a2*b2*n+132*a1^3*a2*n^2+13*a1^3*b1^2*b2-26*a1^3*b1^2*n+44
*a1^3*b1*b2^2-132*a1^3*b1*b2*n+132*a1^3*b1*n^2+33*a1^3*b2^3-132*a1^3*b2^2*n+198
*a1^3*b2*n^2-132*a1^3*n^3+52*a1^2*a2^3*b1+52*a1^2*a2^3*b2-104*a1^2*a2^3*n+96*a1
^2*a2^2*b1^2+320*a1^2*a2^2*b1*b2-679*a1^2*a2^2*b1*n+228*a1^2*a2^2*b2^2-835*a1^2
*a2^2*b2*n+874*a1^2*a2^2*n^2+13*a1^2*a2*b1^3+200*a1^2*a2*b1^2*b2-439*a1^2*a2*b1
^2*n+453*a1^2*a2*b1*b2^2-1829*a1^2*a2*b1*b2*n+1988*a1^2*a2*b1*n^2+270*a1^2*a2*
b2^3-1494*a1^2*a2*b2^2*n+2862*a1^2*a2*b2*n^2-2040*a1^2*a2*n^3+13*a1^2*b1^3*b2-\
26*a1^2*b1^3*n+96*a1^2*b1^2*b2^2-439*a1^2*b1^2*b2*n+478*a1^2*b1^2*n^2+171*a1^2*
b1*b2^3-1098*a1^2*b1*b2^2*n+2268*a1^2*b1*b2*n^2-1644*a1^2*b1*n^3+88*a1^2*b2^4-\
711*a1^2*b2^3*n+2043*a1^2*b2^2*n^2-2664*a1^2*b2*n^3+1431*a1^2*n^4+65*a1*a2^4*b1
+65*a1*a2^4*b2-130*a1*a2^4*n+171*a1*a2^3*b1^2+508*a1*a2^3*b1*b2-1120*a1*a2^3*b1
*n+337*a1*a2^3*b2^2-1380*a1*a2^3*b2*n+1484*a1*a2^3*n^2+44*a1*a2^2*b1^3+453*a1*
a2^2*b1^2*b2-1098*a1*a2^2*b1^2*n+902*a1*a2^2*b1*b2^2-4213*a1*a2^2*b1*b2*n+4892*
a1*a2^2*b1*n^2+493*a1*a2^2*b2^3-3271*a1*a2^2*b2^2*n+7118*a1*a2^2*b2*n^2-5328*a1
*a2^2*n^3+53*a1*a2*b1^3*b2-132*a1*a2*b1^3*n+320*a1*a2*b1^2*b2^2-1829*a1*a2*b1^2
*b2*n+2268*a1*a2*b1^2*n^2+508*a1*a2*b1*b2^3-4213*a1*a2*b1*b2^2*n+10372*a1*a2*b1
*b2*n^2-8240*a1*a2*b1*n^3+241*a1*a2*b2^4-2542*a1*a2*b2^3*n+8978*a1*a2*b2^2*n^2-\
13568*a1*a2*b2*n^3+7804*a1*a2*n^4+13*a1*b1^3*b2^2-106*a1*b1^3*b2*n+132*a1*b1^3*
n^2+52*a1*b1^2*b2^3-679*a1*b1^2*b2^2*n+1988*a1*b1^2*b2*n^2-1644*a1*b1^2*n^3+65*
a1*b1*b2^4-1120*a1*b1*b2^3*n+4892*a1*b1*b2^2*n^2-8240*a1*b1*b2*n^3+4942*a1*b1*n
^4+26*a1*b2^5-547*a1*b2^4*n+3102*a1*b2^3*n^2-7616*a1*b2^2*n^3+8844*a1*b2*n^4-\
4110*a1*n^5+26*a2^5*b1+26*a2^5*b2-52*a2^5*n+88*a2^4*b1^2+241*a2^4*b1*b2-547*a2^
4*b1*n+153*a2^4*b2^2-677*a2^4*b2*n+742*a2^4*n^2+33*a2^3*b1^3+270*a2^3*b1^2*b2-\
711*a2^3*b1^2*n+493*a2^3*b1*b2^2-2542*a2^3*b1*b2*n+3102*a2^3*b1*n^2+256*a2^3*b2
^3-1935*a2^3*b2^2*n+4586*a2^3*b2*n^2-3552*a2^3*n^3+44*a2^2*b1^3*b2-132*a2^2*b1^
3*n+228*a2^2*b1^2*b2^2-1494*a2^2*b1^2*b2*n+2043*a2^2*b1^2*n^2+337*a2^2*b1*b2^3-\
3271*a2^2*b1*b2^2*n+8978*a2^2*b1*b2*n^2-7616*a2^2*b1*n^3+153*a2^2*b2^4-1935*a2^
2*b2^3*n+7809*a2^2*b2^2*n^2-12944*a2^2*b2*n^3+7804*a2^2*n^4+13*a2*b1^3*b2^2-132
*a2*b1^3*b2*n+198*a2*b1^3*n^2+52*a2*b1^2*b2^3-835*a2*b1^2*b2^2*n+2862*a2*b1^2*
b2*n^2-2664*a2*b1^2*n^3+65*a2*b1*b2^4-1380*a2*b1*b2^3*n+7118*a2*b1*b2^2*n^2-\
13568*a2*b1*b2*n^3+8844*a2*b1*n^4+26*a2*b2^5-677*a2*b2^4*n+4586*a2*b2^3*n^2-\
12944*a2*b2^2*n^3+16648*a2*b2*n^4-8220*a2*n^5-26*b1^3*b2^2*n+132*b1^3*b2*n^2-\
132*b1^3*n^3-104*b1^2*b2^3*n+874*b1^2*b2^2*n^2-2040*b1^2*b2*n^3+1431*b1^2*n^4-\
130*b1*b2^4*n+1484*b1*b2^3*n^2-5328*b1*b2^2*n^3+7804*b1*b2*n^4-4110*b1*n^5-52*
b2^5*n+742*b2^4*n^2-3552*b2^3*n^3+7804*b2^2*n^4-8220*b2*n^5+3425*n^6-12*a1^3*a2
^2-51*a1^3*a2*b1-63*a1^3*a2*b2+126*a1^3*a2*n-12*a1^3*b1^2-63*a1^3*b1*b2+126*a1^
3*b1*n-63*a1^3*b2^2+189*a1^3*b2*n-189*a1^3*n^2-48*a1^2*a2^3-330*a1^2*a2^2*b1-\
402*a1^2*a2^2*b2+840*a1^2*a2^2*n-213*a1^2*a2*b1^2-894*a1^2*a2*b1*b2+1941*a1^2*
a2*b1*n-729*a1^2*a2*b2^2+2781*a1^2*a2*b2*n-2970*a1^2*a2*n^2-12*a1^2*b1^3-213*a1
^2*b1^2*b2+462*a1^2*b1^2*n-540*a1^2*b1*b2^2+2214*a1^2*b1*b2*n-2403*a1^2*b1*n^2-\
351*a1^2*b2^3+1998*a1^2*b2^2*n-3888*a1^2*b2*n^2+2781*a1^2*n^3-60*a1*a2^4-546*a1
*a2^3*b1-666*a1*a2^3*b2+1428*a1*a2^3*n-540*a1*a2^2*b1^2-2067*a1*a2^2*b1*b2+4794
*a1*a2^2*b1*n-1599*a1*a2^2*b2^2+6936*a1*a2^2*b2*n-7776*a1*a2^2*n^2-63*a1*a2*b1^
3-894*a1*a2*b1^2*b2+2214*a1*a2*b1^2*n-2067*a1*a2*b1*b2^2+10164*a1*a2*b1*b2*n-\
12105*a1*a2*b1*n^2-1248*a1*a2*b2^3+8790*a1*a2*b2^2*n-19881*a1*a2*b2*n^2+15234*
a1*a2*n^3-51*a1*b1^3*b2+126*a1*b1^3*n-330*a1*b1^2*b2^2+1941*a1*b1^2*b2*n-2403*
a1*b1^2*n^2-546*a1*b1*b2^3+4794*a1*b1*b2^2*n-12105*a1*b1*b2*n^2+9672*a1*b1*n^3-\
267*a1*b2^4+3042*a1*b2^3*n-11187*a1*b2^2*n^2+17289*a1*b2*n^3-10035*a1*n^4-24*a2
^5-267*a2^4*b1-327*a2^4*b2+714*a2^4*n-351*a2^3*b1^2-1248*a2^3*b1*b2+3042*a2^3*
b1*n-945*a2^3*b2^2+4470*a2^3*b2*n-5184*a2^3*n^2-63*a2^2*b1^3-729*a2^2*b1^2*b2+
1998*a2^2*b1^2*n-1599*a2^2*b1*b2^2+8790*a2^2*b1*b2*n-11187*a2^2*b1*n^2-945*a2^2
*b2^3+7632*a2^2*b2^2*n-18963*a2^2*b2*n^2+15234*a2^2*n^3-63*a2*b1^3*b2+189*a2*b1
^3*n-402*a2*b1^2*b2^2+2781*a2*b1^2*b2*n-3888*a2*b1^2*n^2-666*a2*b1*b2^3+6936*a2
*b1*b2^2*n-19881*a2*b1*b2*n^2+17289*a2*b1*n^3-327*a2*b2^4+4470*a2*b2^3*n-18963*
a2*b2^2*n^2+32523*a2*b2*n^3-20070*a2*n^4-12*b1^3*b2^2+126*b1^3*b2*n-189*b1^3*n^
2-48*b1^2*b2^3+840*b1^2*b2^2*n-2970*b1^2*b2*n^2+2781*b1^2*n^3-60*b1*b2^4+1428*
b1*b2^3*n-7776*b1*b2^2*n^2+15234*b1*b2*n^3-10035*b1*n^4-24*b2^5+714*b2^4*n-5184
*b2^3*n^2+15234*b2^2*n^3-20070*b2*n^4+10035*n^5+44*a1^3*a2+44*a1^3*b1+66*a1^3*
b2-132*a1^3*n+296*a1^2*a2^2+702*a1^2*a2*b1+998*a1^2*a2*b2-2128*a1^2*a2*n+164*a1
^2*b1^2+800*a1^2*b1*b2-1732*a1^2*b1*n+724*a1^2*b2^2-2796*a1^2*b2*n+2994*a1^2*n^
2+504*a1*a2^3+1744*a1*a2^2*b1+2500*a1*a2^2*b2-5592*a1*a2^2*n+800*a1*a2*b1^2+
3696*a1*a2*b1*b2-8796*a1*a2*b1*n+3192*a1*a2*b2^2-14388*a1*a2*b2*n+16516*a1*a2*n
^2+44*a1*b1^3+702*a1*b1^2*b2-1732*a1*b1^2*n+1744*a1*b1*b2^2-8796*a1*b1*b2*n+
10528*a1*b1*n^2+1108*a1*b2^3-8128*a1*b2^2*n+18786*a1*b2*n^2-14520*a1*n^3+252*a2
^4+1108*a2^3*b1+1612*a2^3*b2-3728*a2^3*n+724*a2^2*b1^2+3192*a2^2*b1*b2-8128*a2^
2*b1*n+2764*a2^2*b2^2-13720*a2^2*b2*n+16516*a2^2*n^2+66*a2*b1^3+998*a2*b1^2*b2-\
2796*a2*b1^2*n+2500*a2*b1*b2^2-14388*a2*b1*b2*n+18786*a2*b1*n^2+1612*a2*b2^3-\
13720*a2*b2^2*n+35302*a2*b2*n^2-29040*a2*n^3+44*b1^3*b2-132*b1^3*n+296*b1^2*b2^
2-2128*b1^2*b2*n+2994*b1^2*n^2+504*b1*b2^3-5592*b1*b2^2*n+16516*b1*b2*n^2-14520
*b1*n^3+252*b2^4-3728*b2^3*n+16516*b2^2*n^2-29040*b2*n^3+18150*n^4-36*a1^3-600*
a1^2*a2-492*a1^2*b1-792*a1^2*b2+1692*a1^2*n-1584*a1*a2^2-2529*a1*a2*b1-4113*a1*
a2*b2+9426*a1*a2*n-492*a1*b1^2-2529*a1*b1*b2+6042*a1*b1*n-2337*a1*b2^2+10755*a1
*b2*n-12447*a1*n^2-1056*a2^3-2337*a2^2*b1-3921*a2^2*b2+9426*a2^2*n-792*a2*b1^2-\
4113*a2*b1*b2+10755*a2*b1*n-3921*a2*b2^2+20181*a2*b2*n-24894*a2*n^2-36*b1^3-600
*b1^2*b2+1692*b1^2*n-1584*b1*b2^2+9426*b1*b2*n-12447*b1*n^2-1056*b2^3+9426*b2^2
*n-24894*b2*n^2+20745*n^3+392*a1^2+2212*a1*a2+1428*a1*b1+2534*a1*b2-5852*a1*n+
2212*a2^2+2534*a2*b1+4746*a2*b2-11704*a2*n+392*b1^2+2212*b1*b2-5852*b1*n+2212*
b2^2-11704*b2*n+14630*n^2-1164*a1-2328*a2-1164*b1-2328*b2+5820*n+1000)/(2+n)/(
b1+b2-n-2)/(n+2+a1-b1-b2)/(a1+a2-n-2)/(-n-2+a1+a2-b1)/(a1*a2*b1+a1*a2*b2-2*a1*
a2*n+a1*b1*b2-2*a1*b1*n+a1*b2^2-3*a1*b2*n+3*a1*n^2+a2^2*b1+a2^2*b2-2*a2^2*n+a2*
b1*b2-3*a2*b1*n+a2*b2^2-5*a2*b2*n+6*a2*n^2-2*b1*b2*n+3*b1*n^2-2*b2^2*n+6*b2*n^2
-5*n^3-2*a1*a2-2*a1*b1-3*a1*b2+6*a1*n-2*a2^2-3*a2*b1-5*a2*b2+12*a2*n-2*b1*b2+6*
b1*n-2*b2^2+12*b2*n-15*n^2+3*a1+6*a2+3*b1+6*b2-15*n-5)*N+N^2:

add(factor(normal(coeff(ope,N,i)))*N^i,i=0..degree(ope,N)):
end:


#SeqFromRec(ope,n,N,ini,L): Given an (ordinary) recurrence operator
#ope(n,N) in the variable n and the shift operator N (Nf(n):=f(n+1))
#and given initial values [ini0,ini1,..,ini_{ORD-1}],computes
#the first L+1 terms of the sequence a(n) satisfying
#ope(n,N)a(n)=0 and a[i]=ini[i] for i=0,...,ORD-1
#For example, try: SeqFromRec(N^2-N-1,n,N,[0,1],10);

SeqFromRec:=proc(ope,n,N,ini,L)
local gu,lu,j,ORD,n0,i:

ORD:=degree(ope,N):
if ORD<>nops(ini) then
 ERROR(`The length of`,ini, `should be `, degree(ope,N)):
fi:
gu:=ini:
for i from ORD to L do
lu:=0:
n0:=i-ORD:
lu:=0:
for j from 0 to ORD-1 do
lu:=lu+subs(n=n0,coeff(ope,N,j))*gu[n0+j+1]:
od:
lu:=-lu/subs(n=n0,coeff(ope,N,ORD)):
gu:=[op(gu),lu]:
od:
gu:
end:



#IntGBnaked(a1,a2,b1,b2,n1);
#int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1);
#Try:
#IntGBnaked(0,0,0,0,0);
IntGBnaked:=proc(a1,a2,b1,b2,n1) local x,y:
int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n1,x=0..1),y=0..1);
end:

#IntGB(a1,a2,b1,b2,n1);
#int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1);
#Try:
#IntGB(0,0,0,0,0);
IntGB:=proc(a1,a2,b1,b2,n1) local x,y,lu,mu:
lu:=int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n1,x=0..1),y=0..1);
mu:=int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2),x=0..1),y=0..1);
lu/mu:
end:


#IntGBg(a1,a2,b1,b2,c,n1);
#int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1) divided by
#int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^c*(x*(1-x)*y*(1-y))/(1-x*y))^n1,x=0..1),y=0..1) divided by
#Try:
#IntGBg(0,0,0,0,0,0);
IntGBg:=proc(a1,a2,b1,b2,c,n1) local x,y,lu,mu:
lu:=int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n1,x=0..1),y=0..1);
mu:=int(int( x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^c,x=0..1),y=0..1);
lu/mu:
end:


#delt(a,c): Given a rational number a and a constant
#c, finds the delta such that |a-c|=1/denom(a)^(1+delta)
#For example, try delta(22/7,evalf(Pi)):
delt:=proc(a,c): evalf(-log(abs(c-a))/log(denom(a)))-1: end:



#AppxSeq1(ope,n,N,RE,K): It inputs a second order operator ope, in n where N is the shift operator N
#where it is known that ope(n,N) annihilates a sequence a(n) that goes to zero.
#It also inputs a pair RE=[[c0,c1],L] satisfying c0*a(0)+c1*a(1)=L where c0,c1 are integers and
#L is a rational number, It outputs the list of the first K terms in the rational approximations to a(0). Try
#AppxSeq1(N^2-N-1,n,N,[[11,12],3],20);
AppxSeq1:=proc(ope,n,N,RE,K) local c0,c1,L,gu,a0,lu,i,mu0,mu1:

if not (type(RE,list) and nops(RE)=2 and type(RE[1],list) and nops(RE[1])=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:


c0:=RE[1][1]:
c1:=RE[1][2]:

if c1=0 then
  RETURN(FAIL):
fi:

L:=RE[2]:

gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K):

gu:=[op(3..nops(gu),gu)]:
lu:=[]:

for i from 2 to nops(gu) do
 mu0:=coeff(gu[i],a0,0):
 mu1:=coeff(gu[i],a0,1):
 if mu1=0 then
   RETURN(FAIL):
 fi:
 lu:=[op(lu),-mu0/mu1]:
od:
lu:
end:



#AppxSeq(a1,a2,b1,b2,K): The approximating sequence for  IntGB(a1,a2,b1,b2,0); 
#Try:
#AppxSeq(0,0,0,0,40);
#AppxSeq(1/2,0,0,1/2,40);
#AppxSeq(1/3,0,1/3,0,40);
AppxSeq:=proc(a1,a2,b1,b2,K) local ope,n,N,RE:

RE:=FindRel(a1,a2,b1,b2):

if RE=FAIL then
 RETURN(FAIL):
fi:



ope:=OPEZ2(a1,a2,b1,b2,n,N):

AppxSeq1(ope,n,N,RE,K):

end:


#GBC1(a1,a2,b1,b2, K): A floating-point approximation to the constant
#IntGB(a1,a2,b1,b2,0) using the sequence to K terms 
#followed by the differene of two consecutive terms. Try:
#GBC1(0,0,0,0,100);
GBC1:=proc(a1,a2,b1,b2,K)  local RE, n,N,ope,lu:

RE:=FindRel(a1,a2,b1,b2):

if RE=FAIL then
 RETURN(FAIL):
fi:

ope:=OPEZ2(a1,a2,b1,b2,n,N):

lu:=AppxSeq1(ope,n,N,RE,K):

if lu=FAIL then
 RETURN(FAIL):
fi:


[evalf(lu[nops(lu)]),evalf(abs(lu[nops(lu)]-lu[nops(lu)-1]))]:

end:



#GBC(a1,a2,b1,b2): A floating-point approximation to the constant
#IntGB(a1,a2,b1,b2,0) supposed to be good for the number of digits
#Try:
#GBC(0,0,0,0);
GBC:=proc(a1,a2,b1,b2)  local lu,K,RE:


RE:=FindRel(a1,a2,b1,b2):

if RE=FAIL then
 RETURN(FAIL):
fi:

lu:=GBC1(a1,a2,b1,b2,100):

if lu=FAIL then
  RETURN(FAIL):
fi:

if lu[2]<10^(-3*Digits) then
 RETURN(lu[1]):
fi:

for K from 150 to 20000 by 50 do

lu:=GBC1(a1,a2,b1,b2,K):


if lu<>FAIL and lu[2]<10^(-3*Digits) then
 RETURN(lu[1]):
fi:

od:

FAIL:

end:



#deltSeq(a1,a2,b1,b2,K): the sequence of empirial deltas AppxSeq(a1,a2,b1,b2,K) starting at n=10 and until it makes sense
#K must be at least 10.
#to IntGB(a1,a2,b1,b2). Try:
#deltSeq(0,0,0,0,30);
deltSeq:=proc(a1,a2,b1,b2,K) local c,gu,i,lu,RE:

if K<10 then
 print(K, `should have been at least 10`):
 RETURN(FAIL):
fi:

RE:=FindRel(a1,a2,b1,b2):

if RE=FAIL then
  RETURN(FAIL):
fi:

c:=GBC(a1,a2,b1,b2):

if c=FAIL then
 RETURN(FAIL):
fi:
gu:=AppxSeq(a1,a2,b1,b2,K):

lu:=[]:

for i from 11 to nops(gu) while abs(gu[i]-c)>10^(-Digits) do
 lu:=[op(lu),delt(gu[i],c)]:
od:
lu:

end:

#rf(a,n): The raising factorial a(a+1)...*(a+n-1). Try:
#rf(1/3,10);
rf:=proc(a,n) local i:
mul(a+i,i=0..n-1):
end:

#IntGBser(a1,a2,b1,b2,K): an approximation to GBC(a1,a2,b1,b2) using the series with K terms. Try:
#IntGBser(0,0,0,0,100);
#IntGBser(1/2,0,0,1/2,100);
IntGBser:=proc(a1,a2,b1,b2,K) local n1:

add(rf(-a1+1,n1)*rf(-b1+1,n1)/(rf(2-a1-a2,n1)*rf(2-b1-b2,n1)),n1=0..K):

end:



rf1:=proc(a,n): (a+n-1)!/(a-1)!:end:

#IntGBsummand(a1,a2,b1,b2,n): The summand in the series representation for IntGB(a1,a2,b1,b2,0); Try:
#IntGBsummand(0,0,0,0,100);
#IntGBsummand(1/2,0,0,1/2,100);
IntGBsummand:=proc(a1,a2,b1,b2,n):

simplify(rf1(-a1+1,n)*rf1(-b1+1,n)/(rf1(2-a1-a2,n)*rf1(2-b1-b2,n))):

end:


#IntGBsummandRF(a1,a2,b1,b2,n,RF): The summand in the series representation for IntGB(a1,a2,b1,b2,0) in terms of the raising-factorial
#denoted by RF. Try
#IntGBsummandRF(0,0,0,0,n,RF);
#IntGBsummandRF(1/2,0,0,1/2,n,RF);
IntGBsummandRF:=proc(a1,a2,b1,b2,n,RF):

RF(-a1+1,n)*RF(-b1+1,n)/(RF(2-a1-a2,n)*RF(2-b1-b2,n)):

end:



#IntGB3F2(a1,a2,b1,b2): The summand in the series representation for IntGL(a,b,c,0) expressed as an
#2F1, a pair of lists [NumeratorParameters,DenominatorParameters], followed by the argument
#IntGB3F2(0,0,1/2,1/2);
IntGB3F2:=proc(a1,a2,b1,b2) local ku:
ku:=[[1,-a1+1,-b1+1],[2-a1-a2,2-b1-b2]]:
cat(`3F2(` , ku[1][1] , `,` ,  ku[1][2], `,`,  ku[1][3],  `;`,  ku[2][1], `,`, ku[2][2] , `;`, 1, `)`);
end:





#FindRel(a1,a2,b1,b2): A clever way to find a relationship
#between IntGB(a1,a2,b1,b2,0) and  IntGB(a1,a2,b1,b2,1) . Try:
#FindRel(1/2,0,0,1/2);
FindRel:=proc(a1,a2,b1,b2) local x,y,c00,c01,c10, d00,d01,d10,gu,eq,var,lu,A,B,C,gu1,gu2,hal,ka,i,j:

option remember:

lu:=x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2):

gu1:=normal(diff((c00+c01*x+c10*y)*x*(1-x)*lu/(1-x*y),x)+diff((d00+d01*x+d10*y)*y*(1-y)*lu/(1-x*y),y)):

gu1:=normal(gu1/lu):

gu2:=normal(A/(1-x*y)+B*x*(1-x)*y*(1-y)/(1-x*y)^2-C):

gu:=normal(gu1-gu2):

gu:=numer(gu):

var:={c00,c01,c10, d00,d01,d10,A,B,C}:

eq:={seq(seq(coeff(coeff(gu,x,i),y,j),j=0..degree(coeff(gu,x,i),y)),i=0..degree(gu,x))}:

var:=solve(eq,var):

hal:=subs(var,[A,B,C]):

if hal[1]=0 then
 RETURN(FAIL):
fi:

hal:=[1,normal(hal[2]/hal[1]),normal(hal[3]/hal[1])]:

hal:

ka:=lcm(seq(denom(hal[i]),i=1..nops(hal))):

hal:=ka*hal:

[[hal[1],hal[2]],hal[3]]:

end:





#IsGoodOpe(ope,n,N): Is ope(n,N) a good operator?
IsGoodOpe:=proc(ope,n,N) local gu:
gu:=denom(ope):
if subs(n=0,gu)<>0 then
 true:
else
 false:
fi:
end:

#AnBn(a1,a2,b1,b2,K): The first K terms in the sequences An and Bn in the expression
#IntGB(a1,a2,b1,b2,n)=B(n)*c-A(n)
#where c=IntGB(a1,a2,b1,b2,0); 
#It also returns the first K terms in the floating point of the sequence IntGB(a1,a2,b1,b2,n) itself.
#Try:
#AnBn(0,0,0,0,50);
#AnBn(1/2,0,0,1/2,50);
#AnBn(2/3,1/3,1/6,1/3,50);
AnBn:=proc(a1,a2,b1,b2,K) local ope,n,N,RE,c0,c1,L,a0,i,gu,guF:

RE:=FindRel(a1,a2,b1,b2):
if RE=FAIL then
 RETURN(FAIL):
fi:


c0:=RE[1][1]:
c1:=RE[1][2]:

if c1=0 then
  RETURN(FAIL):
fi:

L:=RE[2]:

ope:=OPEZ2(a1,a2,b1,b2,n,N):

if not IsGoodOpe(ope,n,N)  then
 RETURN(FAIL):
fi:

gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K):

guF:=evalf(subs(a0=GBC(a1,a2,b1,b2),gu)):

[[seq(-coeff(gu[i],a0,0),i=1..nops(gu))],[seq(coeff(gu[i],a0,1),i=1..nops(gu))],guF]:

end:





LC:=proc(n) local j:lcm(seq(j,j=1..n)):end:

#Nu1(a,b,K,r): if you have a sequence of diophantine approximations A(n)+B(n)*c such that
#(i)|A(n)+B(n)*c|<=Cb^n, (ii) max(|A(n)|,|B(n)|)=OMEGA(a^n) (iii) K^n*dn(r*n)*A(n) and K^n*dn(r*n)*B(n) are integers
#outputs the implied rigorous delta
#mu:=5*sqrt(5)/2-11/2; 
#Nu1(1/mu,mu,16,4);
#Nu1(1/mu,mu,1,2);
Nu1:=proc(a,b,K,r):
-(log(b)+r+log(K))/(log(a)+r+log(K)):
end:

#Nu1Extra(a,b,K,r,K1): if you have a sequence of diophantine approximations A(n)+B(n)*c such that
#(i)|A(n)+B(n)*c|<=CONST*b^n, (ii) max(|A(n)|,|B(n)|)=OMEGA(a^n) (iii) K^n*dn(r*n)*A(n) and K^n*dn(r*n)*B(n) are integers
#AND it looks like, empirically or rigorously that gcd(K^n*dn(r*n)*A(n), K^n*dn(r*n)*B(n)) are O(e^(K1*n))
#outputs the implied rigorous delta
#Nu1Extra(op(ShoreshPi(N)[2]),25/16,10,0);
Nu1Extra:=proc(a,b,K,r,K1):
-(log(b)+r+log(K)-K1)/(log(a)+r+log(K)-K1):

end:


#GuessK1(L): Given a list of integers finds an integer K such that L[i]/K^i is still an integer or fixed denominator
#Try
#GuessK1([seq(7*5^i,i=1..50)]);

GuessK1:=proc(L) local i,lu,gu,N,K,j,mu:

if nops(L)<40 then
 print(`The list must have at least 40 terms `):
 RETURN(FAIL):
fi:

lu:=ifactors(L[10])[2]:



gu:=[]:

for i from 1 to nops(lu) do
if lu[i][2]>5 then
 gu:=[op(gu),lu[i][1]]:
fi:
od:

if gu=[] then
 RETURN(1):
fi:


N:=nops(L):

mu:=1:

for j from 1 to nops(gu) do

for i from 1 while type(L[N]/gu[j]^i,integer) do od:

K:=round(i/N):

mu:=mu*gu[j]^K:
od:


if max(seq(denom(L[i]/mu^i),i=1..nops(gu)))>100 then
 RETURN(FAIL):
fi:

mu:

end:


#GuessK1(L): Given a list of integers finds an integer K such that L[i]/K^i is still an integer or fixed denominator
#Try
#GuessK1([seq(7*5^i,i=1..50)]);
GuessK:=proc(L) local Lt,Lb,Kt,Kb:

Lt:=numer(L):
Lb:=denom(L):

Kt:=GuessK1(Lt):

if Kt=FAIL then
 RETURN(FAIL):
fi:


Kb:=GuessK1(Lb):

if Kb=FAIL then
 RETURN(FAIL):
fi:

Kb/Kt:

end:



#NoralizePair(P,K): Given a pair of sequenes of rational numbers P[1],P[2], and a pos. rational number K, outputs the
#sequence K^i*P[1][i], K^i*P[2][i]. Try:
#gu:=AnBn(1/2,0,0,1/2,50): Normalize([gu[1],gu[2]],8);
NormalizePair:=proc(P,K) local i:

[[seq(P[1][i]*K^i,i=1..nops(P[1]))],[seq(P[2][i]*K^i,i=1..nops(P[2]))]]:

end:



#CnDn(a1,a2,b1,b2,K): The first K terms in the sequences gcd(numer(An[i]),number(Bn[i]))
#lcm(denom(An[i]),denom(Bn[i]))
#It also returns the last 20 terms of the logs of the normailizing sequence
#as well as the last 20 terms of the implied deltas
#Try:
#CnDn(0,0,0,0,50);
#CnDn(1/2,0,0,1/2,50);
#CnDn(1/2,0,0,1/2,50);
#CnDn(2/3,1/3,1/6,1/3,50);
CnDn:=proc(a1,a2,b1,b2,K) local gu,i,lu,vu,ku:

if K<20 then
 print(K, `shpild be at least 20 `):
 RETURN(FAIL):
fi:

gu:=AnBn(a1,a2,b1,b2,K):

if gu=FAIL then
 RETURN(FAIL):
fi:

lu:=
[
[seq(gcd(numer(gu[1][i]),numer(gu[2][i])),i=1..nops(gu[1]))],
[seq(lcm(denom(gu[1][i]),denom(gu[2][i])),i=1..nops(gu[1]))]
]:

vu:=evalf([seq((log(lu[2][i+1])-log(lu[1][i+1]))/i,i=K-20..K)]):

ku:=[ seq( (log(BC())-vu[i])/(log(BC())+vu[i]),i=1..nops(vu))]:

ku:=evalf(ku,10):
vu:=evalf(vu,10):

[lu,vu,ku]:

end:


#IsGarbage(L): decides whether an expression is garbage. 
IsGarbage:=proc(L) local i,L1:

if not type(L,`*`) then
 RETURN(false):
fi:

for i from 1 to nops(L) do
 if op(0,op(i,L))=int or op(0,op(i,L))=hypergeom then
   RETURN(true):
 fi:
od:

L1:=expand(L):

for i from 1 to nops(L1) do
 if IsGarbage(op(i,L1)) then
   RETURN(true):
 fi:
od:

false:
end:

#hopefulSearchP(numers, denoms): Like
#hopefulSearch(numers, denoms, eps=0.001, blockPosInts=true, report=true) (q.v.)
#but does not print out and also supplies the values of identify(GBC(GoodQuad)) and IntGB at n=0.
# Try:
#       hopefulSearchP([0, 1], [2, 3]);
#       hopefulSearchP([0, 1], [2, 3], eps=0.1);
#       hopefulSearchP([0, 1, 2, 3], [2, 3, 4, 5, 6]);

hopefulSearchP:=proc(numers, denoms) 
local mu,i,gu,ku:
mu:=hopefulSearch(numers, denoms) :
gu:=[]:

for i from 1 to nops(mu) do
ku:=IntGB(op(mu[i][1]),0):
if not IsGarbage(ku) then
 gu:=[op(gu),[mu[i][1],mu[i][2],identify(GBC(op(mu[i][1]))),ku]]:
else
 gu:=[op(gu),[mu[i][1],mu[i][2],identify(GBC(op(mu[i][1])))]]:
fi:
od:

gu:

end:



#LeadA(a1,a2,b1,b2,n): The leading asymptotics in n in the terms of the series IntGBser(a1,a2,b1,b2).
#LeadA(0,0,0,0,n);
LeadA:=proc(a1,a2,b1,b2,n) local lu:

lu:=IntGBsummand(a1,a2,b1,b2,n):

lu:=asympt(lu,n,5);
lu:=simplify(op(1,lu)):

lu:
end:




#FindRelG(a1,a2,b1,b2,c): A clever way to find a relationship
#between IntGBg(a1,a2,b1,b2,c,0) and  IntGBg(a1,a2,b1,b2,c,1) . Try:
#FindRelG(1/2,0,0,1/2,0);
FindRelG:=proc(a1,a2,b1,b2,c) local x,y,c00,c01,c10, d00,d01,d10,gu,eq,var,lu,A,B,C,gu1,gu2,hal,ka,i,j:
option remember:

lu:=x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^c:

gu1:=normal(diff((c00+c01*x+c10*y)*x*(1-x)*lu/(1-x*y),x)+diff((d00+d01*x+d10*y)*y*(1-y)*lu/(1-x*y),y)):

gu1:=normal(gu1/lu):

gu2:=normal(A/(1-x*y)+B*x*(1-x)*y*(1-y)/(1-x*y)^2-C):

gu:=normal(gu1-gu2):

gu:=numer(gu):

var:={c00,c01,c10, d00,d01,d10,A,B,C}:

eq:={seq(seq(coeff(coeff(gu,x,i),y,j),j=0..degree(coeff(gu,x,i),y)),i=0..degree(gu,x))}:

var:=solve(eq,var):

hal:=subs(var,[A,B,C]):

if hal[1]=0 then
 RETURN(FAIL):
fi:

hal:=[1,normal(hal[2]/hal[1]),normal(hal[3]/hal[1])]:

hal:

ka:=lcm(seq(denom(hal[i]),i=1..nops(hal))):

hal:=ka*hal:

[[hal[1],hal[2]],hal[3]]:

end:

#GBCg1(a1,a2,b1,b2,c, K): A floating-point approximation to the constant
#IntGBg(a1,a2,b1,b2,c,0) using the sequence to K terms 
#followed by the differene of two consecutive terms. Try:
#GBCg1(0,0,0,0,100);
GBCg1:=proc(a1,a2,b1,b2,c,K)  local RE, n,N,ope,lu:

RE:=FindRelG(a1,a2,b1,b2,c):

if RE=FAIL then
 RETURN(FAIL):
fi:

ope:=OPEZ2g(a1,a2,b1,b2,c,n,N):

lu:=AppxSeq1(ope,n,N,RE,K):

if lu=FAIL then
 RETURN(FAIL):
fi:


[evalf(lu[nops(lu)]),evalf(abs(lu[nops(lu)]-lu[nops(lu)-1]))]:

end:

#GBCg(a1,a2,b1,b2,c): A floating-point approximation to the constant
#IntGBg(a1,a2,b1,b2,c,0) supposed to be good for the number of digits
#Try:
#GBCg(0,0,0,0,0);
GBCg:=proc(a1,a2,b1,b2,c)  local lu,K,RE:


RE:=FindRelG(a1,a2,b1,b2,c):

if RE=FAIL then
 RETURN(FAIL):
fi:

lu:=GBCg1(a1,a2,b1,b2,c,100):

if lu=FAIL then
  RETURN(FAIL):
fi:

if lu[2]<10^(-3*Digits) then
 RETURN(lu[1]):
fi:

for K from 150 to 20000 by 50 do

lu:=GBCg1(a1,a2,b1,b2,c,K):


if lu<>FAIL and lu[2]<10^(-3*Digits) then
 RETURN(lu[1]):
fi:

od:

FAIL:

end:



#AppxSeqG(a1,a2,b1,b2,c,K): The approximating sequence for  IntGB(a1,a2,b1,b2,0); 
#Try:
#AppxSeqG(0,0,0,0,0,40);
#AppxSeqG(1/2,0,0,1/2,0,40);
#AppxSeqG(1/3,0,1/3,0,0,40);
AppxSeqG:=proc(a1,a2,b1,b2,c,K) local ope,n,N,RE:

RE:=FindRelG(a1,a2,b1,b2,c):

if RE=FAIL then
 RETURN(FAIL):
fi:



ope:=OPEZ2g(a1,a2,b1,b2,c,n,N):

AppxSeq1(ope,n,N,RE,K):

end:




#deltSeqG(a1,a2,b1,b2,c,K): the sequence of empirial deltas AppxSeq(a1,a2,b1,b2,K) starting at n=10 and until it makes sense
#K must be at least 10.
#to IntGBg(a1,a2,b1,b2,c,0). Try:
#deltSeqG(0,0,0,0,0,30);
deltSeqG:=proc(a1,a2,b1,b2,c,K) local c1,gu,i,lu,RE:

if K<10 then
 print(K, `should have been at least 10`):
 RETURN(FAIL):
fi:

RE:=FindRelG(a1,a2,b1,b2,c):

if RE=FAIL then
  RETURN(FAIL):
fi:

c1:=GBCg(a1,a2,b1,b2,c):

if c1=FAIL then
 RETURN(FAIL):
fi:
gu:=AppxSeqG(a1,a2,b1,b2,c,K):

lu:=[]:

for i from 11 to nops(gu) while abs(gu[i]-c1)>10^(-Digits) do
 lu:=[op(lu),delt(gu[i],c1)]:
od:
lu:

end:

#deltSeqGold(a1,a2,b1,b2,c,K): the sequence of empirial deltas AppxSeqG(a1,a2,b1,b2,c,K) starting at n=10 and until it makes sense
#K must be at least 10.
#to IntGBg(a1,a2,b1,b2,c). Try:
#deltSeqG(0,0,0,0,0,30);
deltSeqGold:=proc(a1,a2,b1,b2,c,K) local con1,gu,i,lu,RE:

if K<10 then
 print(K, `should have been at least 10`):
 RETURN(FAIL):
fi:

RE:=FindRelG(a1,a2,b1,b2,c):

if RE=FAIL then
  RETURN(FAIL):
fi:

con1:=GBCg(a1,a2,b1,b2,c):

#I am here

if con1=FAIL then
 RETURN(FAIL):
fi:
gu:=AppxSeqG(a1,a2,b1,b2,c,K):

lu:=[]:

for i from 11 to nops(gu) while abs(gu[i]-c)>10^(-Digits) do
 lu:=[op(lu),delt(gu[i],con1)]:
od:
lu:

end:

 

#PPold(N): Inputs a positive integer N, outputs a list whose entries are
#(i): The largest prime that shows up, let's call it p.
#(ii) the list of lists whose j-th entry is the list primes between sqrt(p) and p with exponent is j.
#Try:
#PPold(LC(100));
PPold:=proc(N) local gu,i,P,gadol,a,T:
gu:=ifactors(N)[2]:

P:=max(seq(gu[i][1],i=1..nops(gu))):


for i from 1 to nops(gu) while gu[i][1]<evalf(sqrt(P)) do od:

a:=i:

gadol:=max(seq(gu[i][2],i=a..nops(gu))):


for i from 1 to gadol do
 T[i]:=[]:
od:

for  i from a to nops(gu) do
 T[gu[i][2]]:=[op(T[gu[i][2]]),gu[i][1]]:
od:

[P, [seq(T[i],i=1..gadol)] ]:

end:



#Khez(n,p): The largest exponent i such that n/p^i is an integer. Try
#Khez(100,5);
Khez:=proc(n,p) local i:

for i from 0 while type(n/p^i,integer) do od:
i-1:
end:







#BC(): The Beukers constant 11/2+5*sqrt(5)/2
BC:=proc(): 11/2+5*sqrt(5)/2:end:




#AnBnG(a1,a2,b1,b2,c,K): The first K terms in the sequences An and Bn in the expression
#IntGBg(a1,a2,b1,b2,c1, n)=B(n)*CONST-A(n)
#where CONST=IntGB(a1,a2,b1,b2,0); 
#It also returns the first K terms in the floating point of the sequence IntGBg(a1,a2,b1,b2,c,n) itself.
#Try:
#AnBnG(0,0,0,0,1,50);
#AnBnG(1/2,0,0,1/2,1,50);
#AnBnG(2/3,1/3,1/6,1/3,1,50);
AnBnG:=proc(a1,a2,b1,b2,c,K) local ope,n,N,RE,c0,c1,L,a0,i,gu,guF:

RE:=FindRelG(a1,a2,b1,b2,c):
if RE=FAIL then
 RETURN(FAIL):
fi:


c0:=RE[1][1]:
c1:=RE[1][2]:

if c1=0 then
  RETURN(FAIL):
fi:

L:=RE[2]:

ope:=OPEZ2g(a1,a2,b1,b2,c,n,N):
gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K):

guF:=evalf(subs(a0=GBCg(a1,a2,b1,b2,c),gu)):

[[seq(-coeff(gu[i],a0,0),i=1..nops(gu))],[seq(coeff(gu[i],a0,1),i=1..nops(gu))],guF]:

end:



#CnDnG(a1,a2,b1,b2,c,K): The first K terms in the sequences gcd(numer(An[i]),number(Bn[i]))
#lcm(denom(An[i]),denom(Bn[i]))
#It also returns the last 20 terms of the logs of the normailizing sequence
#Try:
#CnDnG(0,0,0,0,0,50);
#CnDnG(1/2,0,0,1/2,0,50);
CnDnG:=proc(a1,a2,b1,b2,c,K) local gu,i,lu,ku,vu:

if K<20 then
 print(K, `shpild be at least 20 `):
 RETURN(FAIL):
fi:

gu:=AnBnG(a1,a2,b1,b2,c,K):

lu:=
[
[seq(gcd(numer(gu[1][i]),numer(gu[2][i])),i=1..nops(gu[1]))],
[seq(lcm(denom(gu[1][i]),denom(gu[2][i])),i=1..nops(gu[1]))]
]:


vu:=evalf([seq((log(lu[2][i+1])-log(lu[1][i+1]))/i,i=K-20..K)]):

ku:=[ seq( (log(BC())-vu[i])/(log(BC())+vu[i]),i=1..nops(vu))]:

ku:=evalf(ku,10):
vu:=evalf(vu,10):

[lu,vu,ku]:

end:



###done after I told Robert to modify

#FracPts(n,S,K): Given a large positive integer n, a set of primes S and a pos. const. K
#outputs the set frac(n/p) (in floating point) for all p in S larger than sqrt(K*n). Try:
#FracPts(ithprime(1000),{seq(ithprime(i),i=1..1000)},7);
FracPts:=proc(n,S,K) local gu,s:

gu:={}:

for s in S do
 if s>=evalf(sqrt(K*n)) then
  gu:=gu union {frac(n/s)}:
 fi:
od:

evalf(gu,20):

end:


#HopeStatus(a1,a2,b1,b2,K): inputs a1,a2,b1,b2, such that [a1,a2,b1,b2] is hopeful, checks the
#further rating for the more conserative test of CnDn(a1,a2,b1,b2,K)[3]. If they
#all negative then it returns 0 followed by the smallest neative. If some are negative and some positive, it returns the
#1/2 followed by the smallest negative and larger positive, if they are all positive
#it returns 1 followed by the smallest positive. Try:
#HopeStatus(5/3,5/3,0,0,2000);

HopeStatus:=proc(a1,a2,b1,b2,K) local lu,lu1,lu2,i:
lu:=CnDn(a1,a2,b1,b2,K)[3]:
lu1:={};
lu2:={}:

for i from 1 to nops(lu) do
 if lu[i]<0 then
  lu1:=lu1 union {lu[i]}:
 elif lu[i]>=0 then
  lu2:=lu2 union {lu[i]}:
 fi:
od:

if lu2={} then
 RETURN([0,min(op(lu1))]):
elif lu1={} then
 RETURN([1,min(op(lu2))]):
else
 RETURN([1/2,min(op(lu1)),max(op(lu2))]):
fi:

end:


#TheoremZ2(a1,a2,b1,b2,K,P): Inputs rational numbers a1,a2,b1,b2, and a large positive integer K and outputs
#a theorem regarding the constant
#print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1))/(B(1-a1,1-a2)*B(1-b1,1-b2))
#Either a suggested proof of irrationality or a way to compute it exponentially  fast.
#P is 0 if no INTEGERating factor is known, otherwise it is a proposed one.
#Try:
#TheoremZ2(0,0,0,0,2000,[ [1,0],[1,2],[]] ):
TheoremZ2:=proc(a1,a2,b1,b2,K,P) local n,N,gu,x,y,ope,lu,X,A,B,beta,F,c,E,A1,B1,delta,d1,ka,nuE,deltE,WADIM,su3F2,deltE1:
beta:=BC():
gu:=CnDn(a1,a2,b1,b2,K):
lu:=AnBn(a1,a2,b1,b2,K):
ka:=GBC(a1,a2,b1,b2):
su3F2:=IntGB3F2(a1,a2,b1,b2):

 ope:=OPEZ2(a1,a2,b1,b2,n,N):

print(``):

if min(op(gu[3]))<0 then
 print(`Very fast Computation of rational approximations to the constant `):
 print(cat(`3F2`,`(`, op(su3F2),`;`,1, `)` )):
# print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
 print(`By Shalosh B. Ekhad `):
 print(``):


 print(`Theorem: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates the constant of the title, let's call it c`, cat(`3F2`,`(`, op(su3F2), `;`,1,`)` ) ):
print(`That is easily seen to be given by the following Beukers-type double integral`):
 print(``):
# print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
  print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):


 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):


if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:


if P=0 then

print(`Comment: while this sequence does not lead to an irrationality proof, for the record, the delta, happens to be roughly`):
print(``):
print(min(op(gu[3]))):

print(``):


else
nuE:=AsyPpG(P):
deltE1:= (log(beta)-nuE)/(log(beta)+nuE):
print(`Comment: while this sequence does not lead to an irrationality proof, for the record, the delta, happens to be exactly`):
print(``):
print(deltE1):
print(``):
print(`that in decimals is`):

print(evalf(deltE1,10)):


print(`In order to prove the exact value of the delta such that there exist integer sequences An, Bn such that abs(c-An/Bn)<CONST/Bn^(1+delta)`):
print(`we need the following lemma`):


print(`Lemma : Let E(n) be`):
print(``):
print(PrintPpG(P,n,LCM,PP)):
print(``):
if P[1][2][2]<>0 or P[2][2][2]<>0 then
print(`where LCM(1..m) is the least-common-multiple of the first m integers, and`):
fi:
if P[1][3]<>[] or P[2][3]<>[] then
print(`where for 0<alpha<beta<1, a,b, an integer M and a subset C of {0,...M-1}`):
print(` PP(alpha,beta,a,b,C,M,n) is the product of all r primes between a*n and b*n, such that of you take them mod M it belongs to C`):
print(`and the fractional part of n/p is between alpha and beta`):
fi:
print(``):
print(`Then there exists a sequence G(n) of integers that is asymptotically negligible, i.e. the limit of log( G(n) )/n tends to 0 as n goes to infinity`):
print(`such that `):
 print(` A1(n):=E(n)G(n)A(n), B1(n):=E(n)G(n)B(n) are BOTH integers`):
 print(``):
print(`For example, G(n) for n=`, K, `equals ` ):
WADIM:=lcm(denom(EvalPpG(P,K)*lu[1][-1]),denom(EvalPpG(P,K)*lu[2][-1])):
print(ifactor(WADIM)):
print(`whose log divided by n is`):
print(``):
print(evalf(log(WADIM)/K,10)):
print(``):
print(`Indeed peanuts. `):
print(``):
 print(`Using standard asymptotics derived from the Prime Number Theorem it is readily seen that the limit of`):
print(` log(E(n))/n , and hence log(E(n)*G(n))/n, equals exactly`, nuE ):
 print(``):
print(`that equals in decimals`, evalf(nuE,10)):
print(``):  
 print(`Just as a sanity check, the empircal numerical values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n)*G(n) we get `):
print(``):
print( E(n)*F(n)*G(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nuE)/(log(beta) + nuE) ):
print(``):
print(`that in decimals is, `,  evalf( (log(beta)-nuE)/(log(beta) + nuE) ,10)   )  :
print(``):
print(`Unfortunaley, delta is  negative `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):

fi:



else


 print(`Sketch of an Irrationality Proof of the constant `):
print(cat(`3F2`,`(`, op(su3F2), `;`,1,`)` ) ):

# print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
  print(``):
 print(`By Shalosh B. Ekhad `):
   print(``):
 print(`Theorem: The constant of the title `):
   print(``):
print(cat(`3F2`,`(`, op(su3F2), `;`,1,`)` ) ):
print(``):
print(`let's call it c, that is easily seen to be given by the following Beukers-type integral`):
print(``):
 print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
    print(``):

if P=0 then

 print(`is irrational, with an irrationality measure`, 1+ (log(beta)+nu)/(log(beta)-nu), `for a certain number nu `):
print(`that is approximately `, max(op(gu[2])) , ` yielding an irrationality measure that is approximately `, evalf(1+1/min(op(gu[3])),10)):
print(``):

 print(`We hope that the reader can find  nu exactly. `):

else

nuE:=AsyPpG(P):

deltE:= 1+ (log(beta)+nuE)/(log(beta)-nuE):

 print(`is irrational, with an irrationality measure`, deltE):
print(`that in decimals equals`, evalf( deltE,10) ):
print(``):
fi:

if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

 print(`We need two  lemmas `):
    print(``):

 print(`Lemma ONE: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates the constant of the title`, cat(`3F2`,`(`, op(su3F2), `;`,1,`)`) ):
print(` let's call it c, that can be easily seen to be given by the Beukers-type integral`):
 print(``):
  print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):
 print(`Proof, consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):
print(``):

 print(`we now claim that the sequence of RATIONAL numbers A(n),B(n), can be multiplied by another sequence of rational numbers `):
 print(`E(n) such that both A(n)E(n) and B(n)E(n) are integers `):
 print(``):


if P=0 then
 print(`Lemma TWO: There exists a sequence of rational numbers, whose prime factorizations consists of small primes, that hopefully`):
 print(`can be described (and proved) explicity, that we leave to the expert reader such that `):
 print(` A1(n):=E(n)A(n), B1(n):=E(n)B(n) are BOTH integers`):
 print(``):
 print(`Furthermore there exists a contant, nu, that hopefully the learned reader can determine such that E(n) is OMEGA of `, exp(nu*n) ):
 print(``):
 print(`The empircal values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n) we get `):
print(``):
print( E(n)*F(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nu)/(log(beta) + nu) ):
print(``):
print(`Using the above values of nu for E(n) from`, K+1-nops(gu[3]), `to `, K+1, `the estimated deltas are `):
print(``):
print(gu[3]):
print(``):
print(`As you can see, they are all positive `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):


else

print(`Lemma TWO: Let E(n) be`):
print(``):
print(PrintPpG(P,n,LCM,PP)):
print(``):
if P[1][2][2]<>0 or P[2][2][2]<>0 then
print(`where LCM(1..m) is the least-common-multiple of the first m integers, and`):
fi:
if P[1][3]<>[] or P[2][3]<>[] then
print(`where for 0<alpha<beta<1, a,b, an integer M and a subset C of {0,...M-1}`):
print(` PP(alpha,beta,a,b,C,M,n) is the product of all r primes between a*n and b*n, such that of you take them mod M it belongs to C`):
print(`and the fractional part of n/p is between alpha and beta`):
fi:
print(``):
print(`Then there exists a sequence G(n) of integers that is asymptotically negligible, i.e. the limit of log( G(n) )/n tends to 0 as n goes to infinity`):
print(`such that `):
 print(` A1(n):=E(n)G(n)A(n), B1(n):=E(n)G(n)B(n) are BOTH integers`):
 print(``):
print(`For example, G(n) for n=`, K, `equals ` ):
WADIM:=lcm(denom(EvalPpG(P,K)*lu[1][-1]),denom(EvalPpG(P,K)*lu[2][-1])):
print(ifactor(WADIM)):
print(`whose log divided by n is`):
print(``):
print(evalf(log(WADIM)/K,10)):
print(``):
print(`Indeed peanuts. `):
print(``):
 print(`Using standard asymptotics derived from the Prime Number Theorem it is readily seen that the limit of`):
print(` log(E(n))/n , and hence log(E(n)*G(n))/n, equals exactly`, nuE ):
 print(``):
print(`that equals in decimals`, evalf(nuE,10)):
print(``):  
 print(`Just as a sanity check, the empircal numerical values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n)*G(n) we get `):
print(``):
print( E(n)*F(n)*G(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nuE)/(log(beta) + nuE) ):
print(``):
print(`that in decimals is, `,  evalf( (log(beta)-nuE)/(log(beta) + nuE) ,10)   )  :
print(``):
print(`As you can see, delta is  positive `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):



fi:
fi:

end:



#TheoremZ2num(a1,a2,b1,b2,K,num): Inputs rational numbers a1,a2,b1,b2, and a large positive integer K and outputs
#a theorem regarding the constant
#print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1))/(B(1-a1,1-a2)*B(1-b1,1-b2))
#Either a suggested proof of irrationality or a way to compute it exponentially  fast.
#Try:
#TheoremZ2num(0,0,0,0,2000,2):
TheoremZ2num:=proc(a1,a2,b1,b2,K,num) local n,N,gu,x,y,ope,lu,X,A,B,beta,F,c,E,A1,B1,delta,d1,ka:
beta:=BC():
gu:=CnDn(a1,a2,b1,b2,K):
lu:=AnBn(a1,a2,b1,b2,K):
ka:=GBC(a1,a2,b1,b2):


 ope:=OPEZ2(a1,a2,b1,b2,n,N):

if min(op(gu[3]))<0 then
print(``):
 print(`Theorem Number`, num, `: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates `):
 print(``):
  print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):


 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):


if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

 print(`Proof: Consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):

else


   print(``):

 print(`Theorem number`, num, `: The following constant c. `):
   print(``):
 print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
    print(``):
 print(`is irrational, with an irrationality measure`, 1+ (log(beta)+nu)/(log(beta)-nu), `for a certain number nu `):
print(`that is approximately `, max(op(gu[2])) , ` yielding an irrationality measure that is approximately `, evalf(1+1/min(op(gu[3])),10)):

print(``):
 print(`We hope that the reader can find  nu exactly.`):

if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

 print(`We need two  lemmas `):
    print(``):

 print(`Lemma: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates `):
 print(``):
  print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):
 print(`Proof: Consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of `, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):
print(``):
 print(`we now claim that the sequence of RATIONAL numbers A(n),B(n), can be multiplied by another sequence of rational numbers `):
 print(`E(n) such that both A(n)E(n) and B(n)E(n) are integers `):
 print(``):
 print(`Lemma: There exists a sequence of rational numbers, whose prime factorizations consists of small primes, that hopefully`):
 print(`can be described (and proved) explicity, that we leave to the expert reader such that `):
 print(` A1(n):=E(n)A(n), B1(n):=E(n)B(n) are BOTH integers`):
 print(``):
 print(`Furthermore there exists a contant, nu, that hopefully the learned reader can determine such that E(n) is OMEGA of `, exp(nu*n) ):
 print(``):
 print(`The empircal values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n) we get `):
print(``):
print( E(n)*F(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nu)/(log(beta) + nu) ):
print(``):
print(`Using the above values of nu for E(n) from`, K+1-nops(gu[3]), `to `, K+1, `the estimated deltas are `):
print(``):
print(gu[3]):
print(``):
print(`As you can see, they are all positive `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):
fi:

end:


#PaperZ2(L,K): Given a list L of quadruples that are believed to be provably irrational and a large positive ineger K (around 2000 is OK)
# outputs a paper with sketches
#of proof. Try:
#PaperZ2([[0,0,0,0],[1/2,0,0,0]],2000):
PaperZ2:=proc(L,K) local i,t0:
t0:=time():
print(``):
print(`Sketches of proofs of the Irrationality of `, nops(L), `Constants given as certain double integrals`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for i from 1 to nops(L) do
print(``):
print(`------------------------------------------------------------`):
print(``):
TheoremZ2num(op(L[i]),K,i):
print(``):
od:

print(``):
print(`-----------------------------------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to generate `):
print(``):
end:



#Mamar2: PaperZ2(L,K): where L is given by Hopefuls2(): Try:
#Mamar2(2000):
Mamar2:=proc(K) local L,i:

L:=Hopefuls2():

L:=[seq(L[i][1][1],i=1..nops(L))]:
PaperZ2(L,K):

end:


#Mamar3: PaperZ2(L,K): where L is given by Hopefuls3(): Try:
#Mamar3(2000):
Mamar3:=proc(K) local L,i:

L:=Hopefuls3():

PaperZ2(L,K):

end:


#Mamar4: PaperZ2(L,K): where L is given by Hopefuls3(): Try:
#Mamar4(2000):
Mamar4:=proc(K) local L,i:

L:=Hopefuls4():

PaperZ2(L,K):

end:

# Exhaustively searches a collection of user-supplied arguments for "hopeful"
# values. See hopefulSearch() for an example use.
# tuples: List of "argument generators."
# generateArgs: Function that turns a tuple into a list of arguments.
# blockArgs: Boolean function that possibly tells us to skip some arguments.
# checkRes: Function that returns a numeric "score" for the result of an
#             argument, where bigger and positive is better.
# eps: "Hopeful" tolerance.
# report: If true, print
hopefulSearchGen:=proc(tuples, generateArgs, blockArgs, checkRes,
                        {eps:=0.001, report:=true, tuple_size:=false})
    local blockPosIntMap, hopeful, l, res, new, count:
    local arguments:
    print(tuple_size);

    hopeful := []:

    count := 0;

    for l in tuples do
        count += 1:
        arguments := generateArgs(l):
        if type(tuple_size, integer) then
            printf("%d / %d\n", count, tuple_size);
        fi:

        # Do not retry parameters we've already seen.
        # (e.g, 1 / 2 = 2 / 4)
        if ormap(p -> arguments = p[1], hopeful) then
            next:
        fi:

        if blockArgs(arguments) then
            next:
        fi:

        try
            res := checkRes(arguments):
            if res <> FAIL and res > eps then
                new := [arguments, res]:
                hopeful := [op(hopeful), new]:
                if report then
                    print(new):
                fi:
            fi:
        catch:
            print("error occurred!");
            print(lastexception);
        end:
    od:

    hopeful:
end:

with(Iterator):

# Exhaustively searches a collection of rational parameters to IntGB for
# promising irrationality measures. Searches all possible p / q where p is in
# `numers` and `q` is in `denoms` and reports deltas > `eps`. Optionally blocks
# positive integers (which often results in division by zero errors.)

# Try:
#       hopefulSearch([0, 1], [2, 3]);
#       hopefulSearch([0, 1], [2, 3], eps=0.1);
#       hopefulSearch([0, 1, 2, 3], [2, 3, 4, 5, 6]);
hopefulSearch:=proc(numers, denoms, {eps:=0.001, report:=true})
    local tuples, generateArgs, blockPosIntMap, blockArgs, checkRes:

    tuples := CartesianProduct(numers, denoms, numers, denoms,
                                  numers, denoms, numers, denoms):

    generateArgs := l -> [l[1] / l[2], l[3] / l[4],
                          l[5] / l[6], l[7] / l[8]]:

    blockPosIntMap := x -> x > 0 and type(x, integer):
    blockArgs := L -> ormap(blockPosIntMap, L):
    checkRes := proc(L)
        local res:
        res := deltSeq(seq(L), 50):
        if res = FAIL then
            FAIL
        else
            res[-1]:
        fi:
    end:

    hopefulSearchGen(tuples, generateArgs, blockArgs, checkRes,
                     eps=eps, report=report):
end:

# Exhaustively searches a collection of rational parameters to IntGBg for
# promising irrationality measures.
# Try:

#       hopefulSearch([0, 1], [2, 3], [1, 2], [2, 3]);
hopefulSearchG:=proc(numers, denoms, c_numers, c_denoms, {eps:=0.001, report:=true})
    local tuples, generateArgs, blockPosIntMap, blockArgs, checkRes, n_tuples:

    tuples := CartesianProduct(numers, denoms, numers, denoms,
                                  numers, denoms, numers, denoms,
                                  c_numers, c_denoms):

    n_tuples := (nops(numers) * nops(denoms))^4 * nops(c_numers) * nops(c_denoms):

    generateArgs := l -> [l[1] / l[2], l[3] / l[4],
                          l[5] / l[6], l[7] / l[8],
                          l[9] / l[10]]:

    blockPosIntMap := x -> x > 0 and type(x, integer):
    blockArgs := proc(L)
        if ormap(blockPosIntMap, L) then
            return true:
        fi:

        try
            if type(identify(GBCg(seq(L))), fraction) then
                return true:
            fi:
        catch:
            return true:
        end:

        return false:
    end:

    checkRes := proc(L)
        local res:
        res := deltSeqG(seq(L), 50):
        if res = FAIL then
            FAIL
        else
            res[-1]:
        fi:
    end:

    hopefulSearchGen(tuples, generateArgs, blockArgs, checkRes,
                     eps=eps, report=report, tuple_size=n_tuples):
end:





#CleanUp1(L): Given a list L like in Hopefuls2() kicks out trivial duplicates. Try:
#CleanUp1(Hopefuls2()):
CleanUp1:=proc(L) local lu,i,L1,mu, mu1:

lu:={}:

L1:=[]:

for i from 1 to nops(L) do
 mu:=L[i][1]:
 mu1:=[mu[3],mu[4],mu[1],mu[2]]:

if not member(mu1,lu) then
 L1:=[op(L1),L[i]]:
 lu:=lu union {mu,mu1}:
fi:
od:
L1:
end:


#NorOp(ope,n,N): normalizes the operator ope
NorOp:=proc(ope,N) local i,ope1:

ope1:=ope/coeff(ope,N,degree(ope,N)):
add(factor(coeff(ope1,N,i))*N^i,i=0..degree(ope1,N)):
end:


# Second-order recurrence for the generalized Beukers-Zeta2 integral Int(Int(
# (x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2))/(1-x*y)^(c+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,
# x=0..1), y=0..1).
#
OPEZ2g:= proc(a1,a2,b1,b2,c,n,N) local ope:
ope:=(-2+a1+a2-b1-n)*(-2+a1+a2+c-n)*(-2+b1+b2+c-n\
)*(2+a1-b1-b2+n)*(2+c+n)*(-5+3*a1+6*a2-2*a1*a2-2*a2^2+3*b1-2*a1*b1-3*a2*b1+a1*a2\
*b1+a2^2*b1+6*b2-3*a1*b2-5*a2*b2+a1*a2*b2+a2^2*b2-2*b1*b2+a1*b1*b2+a2*b1*b2-2*b2\
^2+a1*b2^2+a2*b2^2+3*c-a1*c-2*a2*c-b1*c+a2*b1*c-2*b2*c+a1*b2*c+a2*b2*c-15*n+6*a1\
*n+12*a2*n-2*a1*a2*n-2*a2^2*n+6*b1*n-2*a1*b1*n-3*a2*b1*n+12*b2*n-3*a1*b2*n-5*a2*\
b2*n-2*b1*b2*n-2*b2^2*n+6*c*n-a1*c*n-2*a2*c*n-b1*c*n-2*b2*c*n-15*n^2+3*a1*n^2+6*\
a2*n^2+3*b1*n^2+6*b2*n^2+3*c*n^2-5*n^3)*N^2+(1000-1164*a1+392*a1^2-36*a1^3-2328\
*a2+2212*a1*a2-600*a1^2*a2+44*a1^3*a2+2212*a2^2-1584*a1*a2^2+296*a1^2*a2^2-12*a1\
^3*a2^2-1056*a2^3+504*a1*a2^3-48*a1^2*a2^3+252*a2^4-60*a1*a2^4-24*a2^5-1164*b1+1\
428*a1*b1-492*a1^2*b1+44*a1^3*b1+2534*a2*b1-2529*a1*a2*b1+702*a1^2*a2*b1-51*a1^3\
*a2*b1-2337*a2^2*b1+1744*a1*a2^2*b1-330*a1^2*a2^2*b1+13*a1^3*a2^2*b1+1108*a2^3*b\
1-546*a1*a2^3*b1+52*a1^2*a2^3*b1-267*a2^4*b1+65*a1*a2^4*b1+26*a2^5*b1+392*b1^2-4\
92*a1*b1^2+164*a1^2*b1^2-12*a1^3*b1^2-792*a2*b1^2+800*a1*a2*b1^2-213*a1^2*a2*b1^\
2+13*a1^3*a2*b1^2+724*a2^2*b1^2-540*a1*a2^2*b1^2+96*a1^2*a2^2*b1^2-3*a1^3*a2^2*b\
1^2-351*a2^3*b1^2+171*a1*a2^3*b1^2-15*a1^2*a2^3*b1^2+88*a2^4*b1^2-21*a1*a2^4*b1^\
2-9*a2^5*b1^2-36*b1^3+44*a1*b1^3-12*a1^2*b1^3+66*a2*b1^3-63*a1*a2*b1^3+13*a1^2*a\
2*b1^3-63*a2^2*b1^3+44*a1*a2^2*b1^3-6*a1^2*a2^2*b1^3+33*a2^3*b1^3-15*a1*a2^3*b1^\
3+a1^2*a2^3*b1^3-9*a2^4*b1^3+2*a1*a2^4*b1^3+a2^5*b1^3-2328*b2+2534*a1*b2-792*a1^\
2*b2+66*a1^3*b2+4746*a2*b2-4113*a1*a2*b2+998*a1^2*a2*b2-63*a1^3*a2*b2-3921*a2^2*\
b2+2500*a1*a2^2*b2-402*a1^2*a2^2*b2+13*a1^3*a2^2*b2+1612*a2^3*b2-666*a1*a2^3*b2+\
52*a1^2*a2^3*b2-327*a2^4*b2+65*a1*a2^4*b2+26*a2^5*b2+2212*b1*b2-2529*a1*b1*b2+80\
0*a1^2*b1*b2-63*a1^3*b1*b2-4113*a2*b1*b2+3696*a1*a2*b1*b2-894*a1^2*a2*b1*b2+53*a\
1^3*a2*b1*b2+3192*a2^2*b1*b2-2067*a1*a2^2*b1*b2+320*a1^2*a2^2*b1*b2-9*a1^3*a2^2*\
b1*b2-1248*a2^3*b1*b2+508*a1*a2^3*b1*b2-36*a1^2*a2^3*b1*b2+241*a2^4*b1*b2-45*a1*\
a2^4*b1*b2-18*a2^5*b1*b2-600*b1^2*b2+702*a1*b1^2*b2-213*a1^2*b1^2*b2+13*a1^3*b1^\
2*b2+998*a2*b1^2*b2-894*a1*a2*b1^2*b2+200*a1^2*a2*b1^2*b2-9*a1^3*a2*b1^2*b2-729*\
a2^2*b1^2*b2+453*a1*a2^2*b1^2*b2-60*a1^2*a2^2*b1^2*b2+a1^3*a2^2*b1^2*b2+270*a2^3\
*b1^2*b2-99*a1*a2^3*b1^2*b2+5*a1^2*a2^3*b1^2*b2-48*a2^4*b1^2*b2+7*a1*a2^4*b1^2*b\
2+3*a2^5*b1^2*b2+44*b1^3*b2-51*a1*b1^3*b2+13*a1^2*b1^3*b2-63*a2*b1^3*b2+53*a1*a2\
*b1^3*b2-9*a1^2*a2*b1^3*b2+44*a2^2*b1^3*b2-24*a1*a2^2*b1^3*b2+2*a1^2*a2^2*b1^3*b\
2-15*a2^3*b1^3*b2+4*a1*a2^3*b1^3*b2+2*a2^4*b1^3*b2+2212*b2^2-2337*a1*b2^2+724*a1\
^2*b2^2-63*a1^3*b2^2-3921*a2*b2^2+3192*a1*a2*b2^2-729*a1^2*a2*b2^2+44*a1^3*a2*b2\
^2+2764*a2^2*b2^2-1599*a1*a2^2*b2^2+228*a1^2*a2^2*b2^2-6*a1^3*a2^2*b2^2-945*a2^3\
*b2^2+337*a1*a2^3*b2^2-21*a1^2*a2^3*b2^2+153*a2^4*b2^2-24*a1*a2^4*b2^2-9*a2^5*b2\
^2-1584*b1*b2^2+1744*a1*b1*b2^2-540*a1^2*b1*b2^2+44*a1^3*b1*b2^2+2500*a2*b1*b2^2\
-2067*a1*a2*b1*b2^2+453*a1^2*a2*b1*b2^2-24*a1^3*a2*b1*b2^2-1599*a2^2*b1*b2^2+902\
*a1*a2^2*b1*b2^2-114*a1^2*a2^2*b1*b2^2+2*a1^3*a2^2*b1*b2^2+493*a2^3*b1*b2^2-159*\
a1*a2^3*b1*b2^2+7*a1^2*a2^3*b1*b2^2-69*a2^4*b1*b2^2+8*a1*a2^4*b1*b2^2+3*a2^5*b1*\
b2^2+296*b1^2*b2^2-330*a1*b1^2*b2^2+96*a1^2*b1^2*b2^2-6*a1^3*b1^2*b2^2-402*a2*b1\
^2*b2^2+320*a1*a2*b1^2*b2^2-60*a1^2*a2*b1^2*b2^2+2*a1^3*a2*b1^2*b2^2+228*a2^2*b1\
^2*b2^2-114*a1*a2^2*b1^2*b2^2+10*a1^2*a2^2*b1^2*b2^2-60*a2^3*b1^2*b2^2+14*a1*a2^\
3*b1^2*b2^2+6*a2^4*b1^2*b2^2-12*b1^3*b2^2+13*a1*b1^3*b2^2-3*a1^2*b1^3*b2^2+13*a2\
*b1^3*b2^2-9*a1*a2*b1^3*b2^2+a1^2*a2*b1^3*b2^2-6*a2^2*b1^3*b2^2+2*a1*a2^2*b1^3*b\
2^2+a2^3*b1^3*b2^2-1056*b2^3+1108*a1*b2^3-351*a1^2*b2^3+33*a1^3*b2^3+1612*a2*b2^\
3-1248*a1*a2*b2^3+270*a1^2*a2*b2^3-15*a1^3*a2*b2^3-945*a2^2*b2^3+493*a1*a2^2*b2^\
3-60*a1^2*a2^2*b2^3+a1^3*a2^2*b2^3+256*a2^3*b2^3-75*a1*a2^3*b2^3+3*a1^2*a2^3*b2^\
3-30*a2^4*b2^3+3*a1*a2^4*b2^3+a2^5*b2^3+504*b1*b2^3-546*a1*b1*b2^3+171*a1^2*b1*b\
2^3-15*a1^3*b1*b2^3-666*a2*b1*b2^3+508*a1*a2*b1*b2^3-99*a1^2*a2*b1*b2^3+4*a1^3*a\
2*b1*b2^3+337*a2^2*b1*b2^3-159*a1*a2^2*b1*b2^3+14*a1^2*a2^2*b1*b2^3-75*a2^3*b1*b\
2^3+16*a1*a2^3*b1*b2^3+6*a2^4*b1*b2^3-48*b1^2*b2^3+52*a1*b1^2*b2^3-15*a1^2*b1^2*\
b2^3+a1^3*b1^2*b2^3+52*a2*b1^2*b2^3-36*a1*a2*b1^2*b2^3+5*a1^2*a2*b1^2*b2^3-21*a2\
^2*b1^2*b2^3+7*a1*a2^2*b1^2*b2^3+3*a2^3*b1^2*b2^3+252*b2^4-267*a1*b2^4+88*a1^2*b\
2^4-9*a1^3*b2^4-327*a2*b2^4+241*a1*a2*b2^4-48*a1^2*a2*b2^4+2*a1^3*a2*b2^4+153*a2\
^2*b2^4-69*a1*a2^2*b2^4+6*a1^2*a2^2*b2^4-30*a2^3*b2^4+6*a1*a2^3*b2^4+2*a2^4*b2^4\
-60*b1*b2^4+65*a1*b1*b2^4-21*a1^2*b1*b2^4+2*a1^3*b1*b2^4+65*a2*b1*b2^4-45*a1*a2*\
b1*b2^4+7*a1^2*a2*b1*b2^4-24*a2^2*b1*b2^4+8*a1*a2^2*b1*b2^4+3*a2^3*b1*b2^4-24*b2\
^5+26*a1*b2^5-9*a1^2*b2^5+a1^3*b2^5+26*a2*b2^5-18*a1*a2*b2^5+3*a1^2*a2*b2^5-9*a2\
^2*b2^5+3*a1*a2^2*b2^5+a2^3*b2^5-1164*c+1106*a1*c-300*a1^2*c+22*a1^3*c+2212*a2*c\
-1584*a1*a2*c+296*a1^2*a2*c-12*a1^3*a2*c-1584*a2^2*c+756*a1*a2^2*c-72*a1^2*a2^2*\
c+504*a2^3*c-120*a1*a2^3*c-60*a2^4*c+1106*b1*c-984*a1*b1*c+230*a1^2*b1*c-12*a1^3\
*b1*c-2145*a2*b1*c+1392*a1*a2*b1*c-192*a1^2*a2*b1*c-a1^3*a2*b1*c+1580*a2^2*b1*c-\
651*a1*a2^2*b1*c+28*a1^2*a2^2*b1*c+3*a1^3*a2^2*b1*c-522*a2^3*b1*c+94*a1*a2^3*b1*\
c+6*a1^2*a2^3*b1*c+65*a2^4*b1*c+3*a1*a2^4*b1*c-300*b1^2*c+230*a1*b1^2*c-36*a1^2*\
b1^2*c+648*a2*b1^2*c-375*a1*a2*b1^2*c+31*a1^2*a2*b1^2*c+3*a1^3*a2*b1^2*c-513*a2^\
2*b1^2*c+192*a1*a2^2*b1^2*c-3*a1^2*a2^2*b1^2*c-a1^3*a2^2*b1^2*c+181*a2^3*b1^2*c-\
30*a1*a2^3*b1^2*c-2*a1^2*a2^3*b1^2*c-24*a2^4*b1^2*c-a1*a2^4*b1^2*c+22*b1^3*c-12*\
a1*b1^3*c-63*a2*b1^3*c+35*a1*a2*b1^3*c-3*a1^2*a2*b1^3*c+55*a2^2*b1^3*c-21*a1*a2^\
2*b1^3*c+a1^2*a2^2*b1^3*c-21*a2^3*b1^3*c+4*a1*a2^3*b1^3*c+3*a2^4*b1^3*c+2212*b2*\
c-2145*a1*b2*c+648*a1^2*b2*c-63*a1^3*b2*c-3729*a2*b2*c+2688*a1*a2*b2*c-564*a1^2*\
a2*b2*c+35*a1^3*a2*b2*c+2336*a2^2*b2*c-1131*a1*a2^2*b2*c+136*a1^2*a2^2*b2*c-3*a1\
^3*a2^2*b2*c-642*a2^3*b2*c+166*a1*a2^3*b2*c-6*a1^2*a2^3*b2*c+65*a2^4*b2*c-3*a1*a\
2^4*b2*c-1584*b1*b2*c+1392*a1*b1*b2*c-375*a1^2*b1*b2*c+35*a1^3*b1*b2*c+2688*a2*b\
1*b2*c-1680*a1*a2*b1*b2*c+276*a1^2*a2*b1*b2*c-12*a1^3*a2*b1*b2*c-1677*a2^2*b1*b2\
*c+661*a1*a2^2*b1*b2*c-51*a1^2*a2^2*b1*b2*c+456*a2^3*b1*b2*c-84*a1*a2^3*b1*b2*c-\
45*a2^4*b1*b2*c+296*b1^2*b2*c-192*a1*b1^2*b2*c+31*a1^2*b1^2*b2*c-3*a1^3*b1^2*b2*\
c-564*a2*b1^2*b2*c+276*a1*a2*b1^2*b2*c-24*a1^2*a2*b1^2*b2*c+357*a2^2*b1^2*b2*c-1\
08*a1*a2^2*b1^2*b2*c+4*a1^2*a2^2*b1^2*b2*c-93*a2^3*b1^2*b2*c+12*a1*a2^3*b1^2*b2*\
c+8*a2^4*b1^2*b2*c-12*b1^3*b2*c-a1*b1^3*b2*c+3*a1^2*b1^3*b2*c+35*a2*b1^3*b2*c-12\
*a1*a2*b1^3*b2*c-21*a2^2*b1^3*b2*c+4*a1*a2^2*b1^3*b2*c+4*a2^3*b1^3*b2*c-1584*b2^\
2*c+1580*a1*b2^2*c-513*a1^2*b2^2*c+55*a1^3*b2^2*c+2336*a2*b2^2*c-1677*a1*a2*b2^2\
*c+357*a1^2*a2*b2^2*c-21*a1^3*a2*b2^2*c-1236*a2^2*b2^2*c+577*a1*a2^2*b2^2*c-66*a\
1^2*a2^2*b2^2*c+a1^3*a2^2*b2^2*c+275*a2^3*b2^2*c-66*a1*a2^3*b2^2*c+2*a1^2*a2^3*b\
2^2*c-21*a2^4*b2^2*c+a1*a2^4*b2^2*c+756*b1*b2^2*c-651*a1*b1*b2^2*c+192*a1^2*b1*b\
2^2*c-21*a1^3*b1*b2^2*c-1131*a2*b1*b2^2*c+661*a1*a2*b1*b2^2*c-108*a1^2*a2*b1*b2^\
2*c+4*a1^3*a2*b1*b2^2*c+577*a2^2*b1*b2^2*c-198*a1*a2^2*b1*b2^2*c+13*a1^2*a2^2*b1\
*b2^2*c-117*a2^3*b1*b2^2*c+16*a1*a2^3*b1*b2^2*c+7*a2^4*b1*b2^2*c-72*b1^2*b2^2*c+\
28*a1*b1^2*b2^2*c-3*a1^2*b1^2*b2^2*c+a1^3*b1^2*b2^2*c+136*a2*b1^2*b2^2*c-51*a1*a\
2*b1^2*b2^2*c+4*a1^2*a2*b1^2*b2^2*c-66*a2^2*b1^2*b2^2*c+13*a1*a2^2*b1^2*b2^2*c+1\
0*a2^3*b1^2*b2^2*c+3*a1*b1^3*b2^2*c-a1^2*b1^3*b2^2*c-3*a2*b1^3*b2^2*c+a2^2*b1^3*\
b2^2*c+504*b2^3*c-522*a1*b2^3*c+181*a1^2*b2^3*c-21*a1^3*b2^3*c-642*a2*b2^3*c+456\
*a1*a2*b2^3*c-93*a1^2*a2*b2^3*c+4*a1^3*a2*b2^3*c+275*a2^2*b2^3*c-117*a1*a2^2*b2^\
3*c+10*a1^2*a2^2*b2^3*c-45*a2^3*b2^3*c+8*a1*a2^3*b2^3*c+2*a2^4*b2^3*c-120*b1*b2^\
3*c+94*a1*b1*b2^3*c-30*a1^2*b1*b2^3*c+4*a1^3*b1*b2^3*c+166*a2*b1*b2^3*c-84*a1*a2\
*b1*b2^3*c+12*a1^2*a2*b1*b2^3*c-66*a2^2*b1*b2^3*c+16*a1*a2^2*b1*b2^3*c+8*a2^3*b1\
*b2^3*c+6*a1*b1^2*b2^3*c-2*a1^2*b1^2*b2^3*c-6*a2*b1^2*b2^3*c+2*a2^2*b1^2*b2^3*c-\
60*b2^4*c+65*a1*b2^4*c-24*a1^2*b2^4*c+3*a1^3*b2^4*c+65*a2*b2^4*c-45*a1*a2*b2^4*c\
+8*a1^2*a2*b2^4*c-21*a2^2*b2^4*c+7*a1*a2^2*b2^4*c+2*a2^3*b2^4*c+3*a1*b1*b2^4*c-a\
1^2*b1*b2^4*c-3*a2*b1*b2^4*c+a2^2*b1*b2^4*c+392*c^2-300*a1*c^2+88*a1^2*c^2-12*a1\
^3*c^2-600*a2*c^2+296*a1*a2*c^2-48*a1^2*a2*c^2+4*a1^3*a2*c^2+296*a2^2*c^2-72*a1*\
a2^2*c^2+4*a1^2*a2^2*c^2-48*a2^3*c^2-300*b1*c^2+120*a1*b1*c^2-12*a1^2*b1*c^2+4*a\
1^3*b1*c^2+538*a2*b1*c^2-132*a1*a2*b1*c^2-8*a1^2*a2*b1*c^2-294*a2^2*b1*c^2+20*a1\
*a2^2*b1*c^2+6*a1^2*a2^2*b1*c^2+52*a2^3*b1*c^2+6*a1*a2^3*b1*c^2+88*b1^2*c^2-12*a\
1*b1^2*c^2-8*a1^2*b1^2*c^2-186*a2*b1^2*c^2+32*a1*a2*b1^2*c^2+6*a1^2*a2*b1^2*c^2+\
111*a2^2*b1^2*c^2-6*a1*a2^2*b1^2*c^2-2*a1^2*a2^2*b1^2*c^2-21*a2^3*b1^2*c^2-2*a1*\
a2^3*b1^2*c^2-12*b1^3*c^2+4*a1*b1^3*c^2+24*a2*b1^3*c^2-6*a1*a2*b1^3*c^2-15*a2^2*\
b1^3*c^2+2*a1*a2^2*b1^3*c^2+3*a2^3*b1^3*c^2-600*b2*c^2+538*a1*b2*c^2-186*a1^2*b2\
*c^2+24*a1^3*b2*c^2+834*a2*b2*c^2-504*a1*a2*b2*c^2+104*a1^2*a2*b2*c^2-6*a1^3*a2*\
b2*c^2-366*a2^2*b2*c^2+128*a1*a2^2*b2*c^2-12*a1^2*a2^2*b2*c^2+52*a2^3*b2*c^2-6*a\
1*a2^3*b2*c^2+296*b1*b2*c^2-132*a1*b1*b2*c^2+32*a1^2*b1*b2*c^2-6*a1^3*b1*b2*c^2-\
504*a2*b1*b2*c^2+162*a1*a2*b1*b2*c^2-15*a1^2*a2*b1*b2*c^2+242*a2^2*b1*b2*c^2-39*\
a1*a2^2*b1*b2*c^2-36*a2^3*b1*b2*c^2-48*b1^2*b2*c^2-8*a1*b1^2*b2*c^2+6*a1^2*b1^2*\
b2*c^2+104*a2*b1^2*b2*c^2-15*a1*a2*b1^2*b2*c^2-51*a2^2*b1^2*b2*c^2+5*a1*a2^2*b1^\
2*b2*c^2+7*a2^3*b1^2*b2*c^2+4*b1^3*b2*c^2-6*a2*b1^3*b2*c^2+2*a2^2*b1^3*b2*c^2+29\
6*b2^2*c^2-294*a1*b2^2*c^2+111*a1^2*b2^2*c^2-15*a1^3*b2^2*c^2-366*a2*b2^2*c^2+24\
2*a1*a2*b2^2*c^2-51*a1^2*a2*b2^2*c^2+2*a1^3*a2*b2^2*c^2+135*a2^2*b2^2*c^2-51*a1*\
a2^2*b2^2*c^2+4*a1^2*a2^2*b2^2*c^2-15*a2^3*b2^2*c^2+2*a1*a2^3*b2^2*c^2-72*b1*b2^\
2*c^2+20*a1*b1*b2^2*c^2-6*a1^2*b1*b2^2*c^2+2*a1^3*b1*b2^2*c^2+128*a2*b1*b2^2*c^2\
-39*a1*a2*b1*b2^2*c^2+5*a1^2*a2*b1*b2^2*c^2-51*a2^2*b1*b2^2*c^2+8*a1*a2^2*b1*b2^\
2*c^2+5*a2^3*b1*b2^2*c^2+4*b1^2*b2^2*c^2+6*a1*b1^2*b2^2*c^2-2*a1^2*b1^2*b2^2*c^2\
-12*a2*b1^2*b2^2*c^2+4*a2^2*b1^2*b2^2*c^2-48*b2^3*c^2+52*a1*b2^3*c^2-21*a1^2*b2^\
3*c^2+3*a1^3*b2^3*c^2+52*a2*b2^3*c^2-36*a1*a2*b2^3*c^2+7*a1^2*a2*b2^3*c^2-15*a2^\
2*b2^3*c^2+5*a1*a2^2*b2^3*c^2+a2^3*b2^3*c^2+6*a1*b1*b2^3*c^2-2*a1^2*b1*b2^3*c^2-\
6*a2*b1*b2^3*c^2+2*a2^2*b1*b2^3*c^2-36*c^3+22*a1*c^3-12*a1^2*c^3+2*a1^3*c^3+44*a\
2*c^3-12*a1*a2*c^3+4*a1^2*a2*c^3-12*a2^2*c^3+22*b1*c^3+12*a1*b1*c^3-2*a1^2*b1*c^\
3-39*a2*b1*c^3-9*a1*a2*b1*c^3+13*a2^2*b1*c^3+3*a1*a2^2*b1*c^3-12*b1^2*c^3-2*a1*b\
1^2*c^3+18*a2*b1^2*c^3+3*a1*a2*b1^2*c^3-6*a2^2*b1^2*c^3-a1*a2^2*b1^2*c^3+2*b1^3*\
c^3-3*a2*b1^3*c^3+a2^2*b1^3*c^3+44*b2*c^3-39*a1*b2*c^3+18*a1^2*b2*c^3-3*a1^3*b2*\
c^3-51*a2*b2*c^3+27*a1*a2*b2*c^3-6*a1^2*a2*b2*c^3+13*a2^2*b2*c^3-3*a1*a2^2*b2*c^\
3-12*b1*b2*c^3-9*a1*b1*b2*c^3+3*a1^2*b1*b2*c^3+27*a2*b1*b2*c^3-9*a2^2*b1*b2*c^3+\
4*b1^2*b2*c^3-6*a2*b1^2*b2*c^3+2*a2^2*b1^2*b2*c^3-12*b2^2*c^3+13*a1*b2^2*c^3-6*a\
1^2*b2^2*c^3+a1^3*b2^2*c^3+13*a2*b2^2*c^3-9*a1*a2*b2^2*c^3+2*a1^2*a2*b2^2*c^3-3*\
a2^2*b2^2*c^3+a1*a2^2*b2^2*c^3+3*a1*b1*b2^2*c^3-a1^2*b1*b2^2*c^3-3*a2*b1*b2^2*c^\
3+a2^2*b1*b2^2*c^3+5820*n-5852*a1*n+1692*a1^2*n-132*a1^3*n-11704*a2*n+9426*a1*a2\
*n-2128*a1^2*a2*n+126*a1^3*a2*n+9426*a2^2*n-5592*a1*a2^2*n+840*a1^2*a2^2*n-26*a1\
^3*a2^2*n-3728*a2^3*n+1428*a1*a2^3*n-104*a1^2*a2^3*n+714*a2^4*n-130*a1*a2^4*n-52\
*a2^5*n-5852*b1*n+6042*a1*b1*n-1732*a1^2*b1*n+126*a1^3*b1*n+10755*a2*b1*n-8796*a\
1*a2*b1*n+1941*a1^2*a2*b1*n-106*a1^3*a2*b1*n-8128*a2^2*b1*n+4794*a1*a2^2*b1*n-67\
9*a1^2*a2^2*b1*n+18*a1^3*a2^2*b1*n+3042*a2^3*b1*n-1120*a1*a2^3*b1*n+72*a1^2*a2^3\
*b1*n-547*a2^4*b1*n+90*a1*a2^4*b1*n+36*a2^5*b1*n+1692*b1^2*n-1732*a1*b1^2*n+462*\
a1^2*b1^2*n-26*a1^3*b1^2*n-2796*a2*b1^2*n+2214*a1*a2*b1^2*n-439*a1^2*a2*b1^2*n+1\
8*a1^3*a2*b1^2*n+1998*a2^2*b1^2*n-1098*a1*a2^2*b1^2*n+129*a1^2*a2^2*b1^2*n-2*a1^\
3*a2^2*b1^2*n-711*a2^3*b1^2*n+228*a1*a2^3*b1^2*n-10*a1^2*a2^3*b1^2*n+117*a2^4*b1\
^2*n-14*a1*a2^4*b1^2*n-6*a2^5*b1^2*n-132*b1^3*n+126*a1*b1^3*n-26*a1^2*b1^3*n+189\
*a2*b1^3*n-132*a1*a2*b1^3*n+18*a1^2*a2*b1^3*n-132*a2^2*b1^3*n+60*a1*a2^2*b1^3*n-\
4*a1^2*a2^2*b1^3*n+45*a2^3*b1^3*n-10*a1*a2^3*b1^3*n-6*a2^4*b1^3*n-11704*b2*n+107\
55*a1*b2*n-2796*a1^2*b2*n+189*a1^3*b2*n+20181*a2*b2*n-14388*a1*a2*b2*n+2781*a1^2\
*a2*b2*n-132*a1^3*a2*b2*n-13720*a2^2*b2*n+6936*a1*a2^2*b2*n-835*a1^2*a2^2*b2*n+1\
8*a1^3*a2^2*b2*n+4470*a2^3*b2*n-1380*a1*a2^3*b2*n+72*a1^2*a2^3*b2*n-677*a2^4*b2*\
n+90*a1*a2^4*b2*n+36*a2^5*b2*n+9426*b1*b2*n-8796*a1*b1*b2*n+2214*a1^2*b1*b2*n-13\
2*a1^3*b1*b2*n-14388*a2*b1*b2*n+10164*a1*a2*b1*b2*n-1829*a1^2*a2*b1*b2*n+72*a1^3\
*a2*b1*b2*n+8790*a2^2*b1*b2*n-4213*a1*a2^2*b1*b2*n+432*a1^2*a2^2*b1*b2*n-6*a1^3*\
a2^2*b1*b2*n-2542*a2^3*b1*b2*n+684*a1*a2^3*b1*b2*n-24*a1^2*a2^3*b1*b2*n+324*a2^4\
*b1*b2*n-30*a1*a2^4*b1*b2*n-12*a2^5*b1*b2*n-2128*b1^2*b2*n+1941*a1*b1^2*b2*n-439\
*a1^2*b1^2*b2*n+18*a1^3*b1^2*b2*n+2781*a2*b1^2*b2*n-1829*a1*a2*b1^2*b2*n+270*a1^\
2*a2*b1^2*b2*n-6*a1^3*a2*b1^2*b2*n-1494*a2^2*b1^2*b2*n+609*a1*a2^2*b1^2*b2*n-40*\
a1^2*a2^2*b1^2*b2*n+363*a2^3*b1^2*b2*n-66*a1*a2^3*b1^2*b2*n-32*a2^4*b1^2*b2*n+12\
6*b1^3*b2*n-106*a1*b1^3*b2*n+18*a1^2*b1^3*b2*n-132*a2*b1^3*b2*n+72*a1*a2*b1^3*b2\
*n-6*a1^2*a2*b1^3*b2*n+60*a2^2*b1^3*b2*n-16*a1*a2^2*b1^3*b2*n-10*a2^3*b1^3*b2*n+\
9426*b2^2*n-8128*a1*b2^2*n+1998*a1^2*b2^2*n-132*a1^3*b2^2*n-13720*a2*b2^2*n+8790\
*a1*a2*b2^2*n-1494*a1^2*a2*b2^2*n+60*a1^3*a2*b2^2*n+7632*a2^2*b2^2*n-3271*a1*a2^\
2*b2^2*n+309*a1^2*a2^2*b2^2*n-4*a1^3*a2^2*b2^2*n-1935*a2^3*b2^2*n+456*a1*a2^3*b2\
^2*n-14*a1^2*a2^3*b2^2*n+207*a2^4*b2^2*n-16*a1*a2^4*b2^2*n-6*a2^5*b2^2*n-5592*b1\
*b2^2*n+4794*a1*b1*b2^2*n-1098*a1^2*b1*b2^2*n+60*a1^3*b1*b2^2*n+6936*a2*b1*b2^2*\
n-4213*a1*a2*b1*b2^2*n+609*a1^2*a2*b1*b2^2*n-16*a1^3*a2*b1*b2^2*n-3271*a2^2*b1*b\
2^2*n+1212*a1*a2^2*b1*b2^2*n-76*a1^2*a2^2*b1*b2^2*n+663*a2^3*b1*b2^2*n-106*a1*a2\
^3*b1*b2^2*n-46*a2^4*b1*b2^2*n+840*b1^2*b2^2*n-679*a1*b1^2*b2^2*n+129*a1^2*b1^2*\
b2^2*n-4*a1^3*b1^2*b2^2*n-835*a2*b1^2*b2^2*n+432*a1*a2*b1^2*b2^2*n-40*a1^2*a2*b1\
^2*b2^2*n+309*a2^2*b1^2*b2^2*n-76*a1*a2^2*b1^2*b2^2*n-40*a2^3*b1^2*b2^2*n-26*b1^\
3*b2^2*n+18*a1*b1^3*b2^2*n-2*a1^2*b1^3*b2^2*n+18*a2*b1^3*b2^2*n-6*a1*a2*b1^3*b2^\
2*n-4*a2^2*b1^3*b2^2*n-3728*b2^3*n+3042*a1*b2^3*n-711*a1^2*b2^3*n+45*a1^3*b2^3*n\
+4470*a2*b2^3*n-2542*a1*a2*b2^3*n+363*a1^2*a2*b2^3*n-10*a1^3*a2*b2^3*n-1935*a2^2\
*b2^3*n+663*a1*a2^2*b2^3*n-40*a1^2*a2^2*b2^3*n+345*a2^3*b2^3*n-50*a1*a2^3*b2^3*n\
-20*a2^4*b2^3*n+1428*b1*b2^3*n-1120*a1*b1*b2^3*n+228*a1^2*b1*b2^3*n-10*a1^3*b1*b\
2^3*n-1380*a2*b1*b2^3*n+684*a1*a2*b1*b2^3*n-66*a1^2*a2*b1*b2^3*n+456*a2^2*b1*b2^\
3*n-106*a1*a2^2*b1*b2^3*n-50*a2^3*b1*b2^3*n-104*b1^2*b2^3*n+72*a1*b1^2*b2^3*n-10\
*a1^2*b1^2*b2^3*n+72*a2*b1^2*b2^3*n-24*a1*a2*b1^2*b2^3*n-14*a2^2*b1^2*b2^3*n+714\
*b2^4*n-547*a1*b2^4*n+117*a1^2*b2^4*n-6*a1^3*b2^4*n-677*a2*b2^4*n+324*a1*a2*b2^4\
*n-32*a1^2*a2*b2^4*n+207*a2^2*b2^4*n-46*a1*a2^2*b2^4*n-20*a2^3*b2^4*n-130*b1*b2^\
4*n+90*a1*b1*b2^4*n-14*a1^2*b1*b2^4*n+90*a2*b1*b2^4*n-30*a1*a2*b1*b2^4*n-16*a2^2\
*b1*b2^4*n-52*b2^5*n+36*a1*b2^5*n-6*a1^2*b2^5*n+36*a2*b2^5*n-12*a1*a2*b2^5*n-6*a\
2^2*b2^5*n-5852*c*n+4713*a1*c*n-1064*a1^2*c*n+63*a1^3*c*n+9426*a2*c*n-5592*a1*a2\
*c*n+840*a1^2*a2*c*n-26*a1^3*a2*c*n-5592*a2^2*c*n+2142*a1*a2^2*c*n-156*a1^2*a2^2\
*c*n+1428*a2^3*c*n-260*a1*a2^3*c*n-130*a2^4*c*n+4713*b1*c*n-3464*a1*b1*c*n+651*a\
1^2*b1*c*n-26*a1^3*b1*c*n-7460*a2*b1*c*n+3852*a1*a2*b1*c*n-407*a1^2*a2*b1*c*n+43\
32*a2^2*b1*c*n-1343*a1*a2^2*b1*c*n+42*a1^2*a2^2*b1*c*n+2*a1^3*a2^2*b1*c*n-1068*a\
2^3*b1*c*n+132*a1*a2^3*b1*c*n+4*a1^2*a2^3*b1*c*n+90*a2^4*b1*c*n+2*a1*a2^4*b1*c*n\
-1064*b1^2*c*n+651*a1*b1^2*c*n-78*a1^2*b1^2*c*n+1782*a2*b1^2*c*n-763*a1*a2*b1^2*\
c*n+42*a1^2*a2*b1^2*c*n+2*a1^3*a2*b1^2*c*n-1035*a2^2*b1^2*c*n+255*a1*a2^2*b1^2*c\
*n-2*a1^2*a2^2*b1^2*c*n+240*a2^3*b1^2*c*n-20*a1*a2^3*b1^2*c*n-16*a2^4*b1^2*c*n+6\
3*b1^3*c*n-26*a1*b1^3*c*n-132*a2*b1^3*c*n+48*a1*a2*b1^3*c*n-2*a1^2*a2*b1^3*c*n+7\
5*a2^2*b1^3*c*n-14*a1*a2^2*b1^3*c*n-14*a2^3*b1^3*c*n+9426*b2*c*n-7460*a1*b2*c*n+\
1782*a1^2*b2*c*n-132*a1^3*b2*c*n-13052*a2*b2*c*n+7416*a1*a2*b2*c*n-1159*a1^2*a2*\
b2*c*n+48*a1^3*a2*b2*c*n+6474*a2^2*b2*c*n-2329*a1*a2^2*b2*c*n+186*a1^2*a2^2*b2*c\
*n-2*a1^3*a2^2*b2*c*n-1328*a2^3*b2*c*n+228*a1*a2^3*b2*c*n-4*a1^2*a2^3*b2*c*n+90*\
a2^4*b2*c*n-2*a1*a2^4*b2*c*n-5592*b1*b2*c*n+3852*a1*b1*b2*c*n-763*a1^2*b1*b2*c*n\
+48*a1^3*b1*b2*c*n+7416*a2*b1*b2*c*n-3424*a1*a2*b1*b2*c*n+372*a1^2*a2*b1*b2*c*n-\
8*a1^3*a2*b1*b2*c*n-3413*a2^2*b1*b2*c*n+888*a1*a2^2*b1*b2*c*n-34*a1^2*a2^2*b1*b2\
*c*n+612*a2^3*b1*b2*c*n-56*a1*a2^3*b1*b2*c*n-30*a2^4*b1*b2*c*n+840*b1^2*b2*c*n-4\
07*a1*b1^2*b2*c*n+42*a1^2*b1^2*b2*c*n-2*a1^3*b1^2*b2*c*n-1159*a2*b1^2*b2*c*n+372\
*a1*a2*b1^2*b2*c*n-16*a1^2*a2*b1^2*b2*c*n+480*a2^2*b1^2*b2*c*n-72*a1*a2^2*b1^2*b\
2*c*n-62*a2^3*b1^2*b2*c*n-26*b1^3*b2*c*n+2*a1^2*b1^3*b2*c*n+48*a2*b1^3*b2*c*n-8*\
a1*a2*b1^3*b2*c*n-14*a2^2*b1^3*b2*c*n-5592*b2^2*c*n+4332*a1*b2^2*c*n-1035*a1^2*b\
2^2*c*n+75*a1^3*b2^2*c*n+6474*a2*b2^2*c*n-3413*a1*a2*b2^2*c*n+480*a1^2*a2*b2^2*c\
*n-14*a1^3*a2*b2^2*c*n-2534*a2^2*b2^2*c*n+777*a1*a2^2*b2^2*c*n-44*a1^2*a2^2*b2^2\
*c*n+372*a2^3*b2^2*c*n-44*a1*a2^3*b2^2*c*n-14*a2^4*b2^2*c*n+2142*b1*b2^2*c*n-134\
3*a1*b1*b2^2*c*n+255*a1^2*b1*b2^2*c*n-14*a1^3*b1*b2^2*c*n-2329*a2*b1*b2^2*c*n+88\
8*a1*a2*b1*b2^2*c*n-72*a1^2*a2*b1*b2^2*c*n+777*a2^2*b1*b2^2*c*n-132*a1*a2^2*b1*b\
2^2*c*n-78*a2^3*b1*b2^2*c*n-156*b1^2*b2^2*c*n+42*a1*b1^2*b2^2*c*n-2*a1^2*b1^2*b2\
^2*c*n+186*a2*b1^2*b2^2*c*n-34*a1*a2*b1^2*b2^2*c*n-44*a2^2*b1^2*b2^2*c*n+2*a1*b1\
^3*b2^2*c*n-2*a2*b1^3*b2^2*c*n+1428*b2^3*c*n-1068*a1*b2^3*c*n+240*a1^2*b2^3*c*n-\
14*a1^3*b2^3*c*n-1328*a2*b2^3*c*n+612*a1*a2*b2^3*c*n-62*a1^2*a2*b2^3*c*n+372*a2^\
2*b2^3*c*n-78*a1*a2^2*b2^3*c*n-30*a2^3*b2^3*c*n-260*b1*b2^3*c*n+132*a1*b1*b2^3*c\
*n-20*a1^2*b1*b2^3*c*n+228*a2*b1*b2^3*c*n-56*a1*a2*b1*b2^3*c*n-44*a2^2*b1*b2^3*c\
*n+4*a1*b1^2*b2^3*c*n-4*a2*b1^2*b2^3*c*n-130*b2^4*c*n+90*a1*b2^4*c*n-16*a1^2*b2^\
4*c*n+90*a2*b2^4*c*n-30*a1*a2*b2^4*c*n-14*a2^2*b2^4*c*n+2*a1*b1*b2^4*c*n-2*a2*b1\
*b2^4*c*n+1692*c^2*n-1064*a1*c^2*n+246*a1^2*c^2*n-26*a1^3*c^2*n-2128*a2*c^2*n+84\
0*a1*a2*c^2*n-104*a1^2*a2*c^2*n+6*a1^3*a2*c^2*n+840*a2^2*c^2*n-156*a1*a2^2*c^2*n\
+6*a1^2*a2^2*c^2*n-104*a2^3*c^2*n-1064*b1*c^2*n+348*a1*b1*c^2*n-26*a1^2*b1*c^2*n\
+6*a1^3*b1*c^2*n+1479*a2*b1*c^2*n-277*a1*a2*b1*c^2*n-9*a1^2*a2*b1*c^2*n-601*a2^2\
*b1*c^2*n+30*a1*a2^2*b1*c^2*n+4*a1^2*a2^2*b1*c^2*n+72*a2^3*b1*c^2*n+4*a1*a2^3*b1\
*c^2*n+246*b1^2*c^2*n-26*a1*b1^2*c^2*n-12*a1^2*b1^2*c^2*n-376*a2*b1^2*c^2*n+42*a\
1*a2*b1^2*c^2*n+4*a1^2*a2*b1^2*c^2*n+147*a2^2*b1^2*c^2*n-4*a1*a2^2*b1^2*c^2*n-14\
*a2^3*b1^2*c^2*n-26*b1^3*c^2*n+6*a1*b1^3*c^2*n+33*a2*b1^3*c^2*n-4*a1*a2*b1^3*c^2\
*n-10*a2^2*b1^3*c^2*n-2128*b2*c^2*n+1479*a1*b2*c^2*n-376*a1^2*b2*c^2*n+33*a1^3*b\
2*c^2*n+2319*a2*b2*c^2*n-1029*a1*a2*b2*c^2*n+141*a1^2*a2*b2*c^2*n-4*a1^3*a2*b2*c\
^2*n-757*a2^2*b2*c^2*n+174*a1*a2^2*b2*c^2*n-8*a1^2*a2^2*b2*c^2*n+72*a2^3*b2*c^2*\
n-4*a1*a2^3*b2*c^2*n+840*b1*b2*c^2*n-277*a1*b1*b2*c^2*n+42*a1^2*b1*b2*c^2*n-4*a1\
^3*b1*b2*c^2*n-1029*a2*b1*b2*c^2*n+216*a1*a2*b1*b2*c^2*n-10*a1^2*a2*b1*b2*c^2*n+\
324*a2^2*b1*b2*c^2*n-26*a1*a2^2*b1*b2*c^2*n-24*a2^3*b1*b2*c^2*n-104*b1^2*b2*c^2*\
n-9*a1*b1^2*b2*c^2*n+4*a1^2*b1^2*b2*c^2*n+141*a2*b1^2*b2*c^2*n-10*a1*a2*b1^2*b2*\
c^2*n-34*a2^2*b1^2*b2*c^2*n+6*b1^3*b2*c^2*n-4*a2*b1^3*b2*c^2*n+840*b2^2*c^2*n-60\
1*a1*b2^2*c^2*n+147*a1^2*b2^2*c^2*n-10*a1^3*b2^2*c^2*n-757*a2*b2^2*c^2*n+324*a1*\
a2*b2^2*c^2*n-34*a1^2*a2*b2^2*c^2*n+183*a2^2*b2^2*c^2*n-34*a1*a2^2*b2^2*c^2*n-10\
*a2^3*b2^2*c^2*n-156*b1*b2^2*c^2*n+30*a1*b1*b2^2*c^2*n-4*a1^2*b1*b2^2*c^2*n+174*\
a2*b1*b2^2*c^2*n-26*a1*a2*b1*b2^2*c^2*n-34*a2^2*b1*b2^2*c^2*n+6*b1^2*b2^2*c^2*n+\
4*a1*b1^2*b2^2*c^2*n-8*a2*b1^2*b2^2*c^2*n-104*b2^3*c^2*n+72*a1*b2^3*c^2*n-14*a1^\
2*b2^3*c^2*n+72*a2*b2^3*c^2*n-24*a1*a2*b2^3*c^2*n-10*a2^2*b2^3*c^2*n+4*a1*b1*b2^\
3*c^2*n-4*a2*b1*b2^3*c^2*n-132*c^3*n+63*a1*c^3*n-26*a1^2*c^3*n+3*a1^3*c^3*n+126*\
a2*c^3*n-26*a1*a2*c^3*n+6*a1^2*a2*c^3*n-26*a2^2*c^3*n+63*b1*c^3*n+26*a1*b1*c^3*n\
-3*a1^2*b1*c^3*n-80*a2*b1*c^3*n-12*a1*a2*b1*c^3*n+18*a2^2*b1*c^3*n+2*a1*a2^2*b1*\
c^3*n-26*b1^2*c^3*n-3*a1*b1^2*c^3*n+24*a2*b1^2*c^3*n+2*a1*a2*b1^2*c^3*n-4*a2^2*b\
1^2*c^3*n+3*b1^3*c^3*n-2*a2*b1^3*c^3*n+126*b2*c^3*n-80*a1*b2*c^3*n+24*a1^2*b2*c^\
3*n-2*a1^3*b2*c^3*n-106*a2*b2*c^3*n+36*a1*a2*b2*c^3*n-4*a1^2*a2*b2*c^3*n+18*a2^2\
*b2*c^3*n-2*a1*a2^2*b2*c^3*n-26*b1*b2*c^3*n-12*a1*b1*b2*c^3*n+2*a1^2*b1*b2*c^3*n\
+36*a2*b1*b2*c^3*n-6*a2^2*b1*b2*c^3*n+6*b1^2*b2*c^3*n-4*a2*b1^2*b2*c^3*n-26*b2^2\
*c^3*n+18*a1*b2^2*c^3*n-4*a1^2*b2^2*c^3*n+18*a2*b2^2*c^3*n-6*a1*a2*b2^2*c^3*n-2*\
a2^2*b2^2*c^3*n+2*a1*b1*b2^2*c^3*n-2*a2*b1*b2^2*c^3*n+14630*n^2-12447*a1*n^2+299\
4*a1^2*n^2-189*a1^3*n^2-24894*a2*n^2+16516*a1*a2*n^2-2970*a1^2*a2*n^2+132*a1^3*a\
2*n^2+16516*a2^2*n^2-7776*a1*a2^2*n^2+874*a1^2*a2^2*n^2-18*a1^3*a2^2*n^2-5184*a2\
^3*n^2+1484*a1*a2^3*n^2-72*a1^2*a2^3*n^2+742*a2^4*n^2-90*a1*a2^4*n^2-36*a2^5*n^2\
-12447*b1*n^2+10528*a1*b1*n^2-2403*a1^2*b1*n^2+132*a1^3*b1*n^2+18786*a2*b1*n^2-1\
2105*a1*a2*b1*n^2+1988*a1^2*a2*b1*n^2-72*a1^3*a2*b1*n^2-11187*a2^2*b1*n^2+4892*a\
1*a2^2*b1*n^2-459*a1^2*a2^2*b1*n^2+6*a1^3*a2^2*b1*n^2+3102*a2^3*b1*n^2-756*a1*a2\
^3*b1*n^2+24*a1^2*a2^3*b1*n^2-369*a2^4*b1*n^2+30*a1*a2^4*b1*n^2+12*a2^5*b1*n^2+2\
994*b1^2*n^2-2403*a1*b1^2*n^2+478*a1^2*b1^2*n^2-18*a1^3*b1^2*n^2-3888*a2*b1^2*n^\
2+2268*a1*a2*b1^2*n^2-297*a1^2*a2*b1^2*n^2+6*a1^3*a2*b1^2*n^2+2043*a2^2*b1^2*n^2\
-738*a1*a2^2*b1^2*n^2+43*a1^2*a2^2*b1^2*n^2-477*a2^3*b1^2*n^2+76*a1*a2^3*b1^2*n^\
2+39*a2^4*b1^2*n^2-189*b1^3*n^2+132*a1*b1^3*n^2-18*a1^2*b1^3*n^2+198*a2*b1^3*n^2\
-90*a1*a2*b1^3*n^2+6*a1^2*a2*b1^3*n^2-90*a2^2*b1^3*n^2+20*a1*a2^2*b1^3*n^2+15*a2\
^3*b1^3*n^2-24894*b2*n^2+18786*a1*b2*n^2-3888*a1^2*b2*n^2+198*a1^3*b2*n^2+35302*\
a2*b2*n^2-19881*a1*a2*b2*n^2+2862*a1^2*a2*b2*n^2-90*a1^3*a2*b2*n^2-18963*a2^2*b2\
*n^2+7118*a1*a2^2*b2*n^2-567*a1^2*a2^2*b2*n^2+6*a1^3*a2^2*b2*n^2+4586*a2^3*b2*n^\
2-936*a1*a2^3*b2*n^2+24*a1^2*a2^3*b2*n^2-459*a2^4*b2*n^2+30*a1*a2^4*b2*n^2+12*a2\
^5*b2*n^2+16516*b1*b2*n^2-12105*a1*b1*b2*n^2+2268*a1^2*b1*b2*n^2-90*a1^3*b1*b2*n\
^2-19881*a2*b1*b2*n^2+10372*a1*a2*b1*b2*n^2-1233*a1^2*a2*b1*b2*n^2+24*a1^3*a2*b1\
*b2*n^2+8978*a2^2*b1*b2*n^2-2835*a1*a2^2*b1*b2*n^2+144*a1^2*a2^2*b1*b2*n^2-1710*\
a2^3*b1*b2*n^2+228*a1*a2^3*b1*b2*n^2+108*a2^4*b1*b2*n^2-2970*b1^2*b2*n^2+1988*a1\
*b1^2*b2*n^2-297*a1^2*b1^2*b2*n^2+6*a1^3*b1^2*b2*n^2+2862*a2*b1^2*b2*n^2-1233*a1\
*a2*b1^2*b2*n^2+90*a1^2*a2*b1^2*b2*n^2-1008*a2^2*b1^2*b2*n^2+203*a1*a2^2*b1^2*b2\
*n^2+121*a2^3*b1^2*b2*n^2+132*b1^3*b2*n^2-72*a1*b1^3*b2*n^2+6*a1^2*b1^3*b2*n^2-9\
0*a2*b1^3*b2*n^2+24*a1*a2*b1^3*b2*n^2+20*a2^2*b1^3*b2*n^2+16516*b2^2*n^2-11187*a\
1*b2^2*n^2+2043*a1^2*b2^2*n^2-90*a1^3*b2^2*n^2-18963*a2*b2^2*n^2+8978*a1*a2*b2^2\
*n^2-1008*a1^2*a2*b2^2*n^2+20*a1^3*a2*b2^2*n^2+7809*a2^2*b2^2*n^2-2205*a1*a2^2*b\
2^2*n^2+103*a1^2*a2^2*b2^2*n^2-1305*a2^3*b2^2*n^2+152*a1*a2^3*b2^2*n^2+69*a2^4*b\
2^2*n^2-7776*b1*b2^2*n^2+4892*a1*b1*b2^2*n^2-738*a1^2*b1*b2^2*n^2+20*a1^3*b1*b2^\
2*n^2+7118*a2*b1*b2^2*n^2-2835*a1*a2*b1*b2^2*n^2+203*a1^2*a2*b1*b2^2*n^2-2205*a2\
^2*b1*b2^2*n^2+404*a1*a2^2*b1*b2^2*n^2+221*a2^3*b1*b2^2*n^2+874*b1^2*b2^2*n^2-45\
9*a1*b1^2*b2^2*n^2+43*a1^2*b1^2*b2^2*n^2-567*a2*b1^2*b2^2*n^2+144*a1*a2*b1^2*b2^\
2*n^2+103*a2^2*b1^2*b2^2*n^2-18*b1^3*b2^2*n^2+6*a1*b1^3*b2^2*n^2+6*a2*b1^3*b2^2*\
n^2-5184*b2^3*n^2+3102*a1*b2^3*n^2-477*a1^2*b2^3*n^2+15*a1^3*b2^3*n^2+4586*a2*b2\
^3*n^2-1710*a1*a2*b2^3*n^2+121*a1^2*a2*b2^3*n^2-1305*a2^2*b2^3*n^2+221*a1*a2^2*b\
2^3*n^2+115*a2^3*b2^3*n^2+1484*b1*b2^3*n^2-756*a1*b1*b2^3*n^2+76*a1^2*b1*b2^3*n^\
2-936*a2*b1*b2^3*n^2+228*a1*a2*b1*b2^3*n^2+152*a2^2*b1*b2^3*n^2-72*b1^2*b2^3*n^2\
+24*a1*b1^2*b2^3*n^2+24*a2*b1^2*b2^3*n^2+742*b2^4*n^2-369*a1*b2^4*n^2+39*a1^2*b2\
^4*n^2-459*a2*b2^4*n^2+108*a1*a2*b2^4*n^2+69*a2^2*b2^4*n^2-90*b1*b2^4*n^2+30*a1*\
b1*b2^4*n^2+30*a2*b1*b2^4*n^2-36*b2^5*n^2+12*a1*b2^5*n^2+12*a2*b2^5*n^2-12447*c*\
n^2+8258*a1*c*n^2-1485*a1^2*c*n^2+66*a1^3*c*n^2+16516*a2*c*n^2-7776*a1*a2*c*n^2+\
874*a1^2*a2*c*n^2-18*a1^3*a2*c*n^2-7776*a2^2*c*n^2+2226*a1*a2^2*c*n^2-108*a1^2*a\
2^2*c*n^2+1484*a2^3*c*n^2-180*a1*a2^3*c*n^2-90*a2^4*c*n^2+8258*b1*c*n^2-4806*a1*\
b1*c*n^2+676*a1^2*b1*c*n^2-18*a1^3*b1*c*n^2-10269*a2*b1*c*n^2+3948*a1*a2*b1*c*n^\
2-279*a1^2*a2*b1*c*n^2+4414*a2^2*b1*c*n^2-909*a1*a2^2*b1*c*n^2+14*a1^2*a2^2*b1*c\
*n^2-720*a2^3*b1*c*n^2+44*a1*a2^3*b1*c*n^2+30*a2^4*b1*c*n^2-1485*b1^2*c*n^2+676*\
a1*b1^2*c*n^2-54*a1^2*b1^2*c*n^2+1818*a2*b1^2*c*n^2-513*a1*a2*b1^2*c*n^2+14*a1^2\
*a2*b1^2*c*n^2-693*a2^2*b1^2*c*n^2+85*a1*a2^2*b1^2*c*n^2+80*a2^3*b1^2*c*n^2+66*b\
1^3*c*n^2-18*a1*b1^3*c*n^2-90*a2*b1^3*c*n^2+16*a1*a2*b1^3*c*n^2+25*a2^2*b1^3*c*n\
^2+16516*b2*c*n^2-10269*a1*b2*c*n^2+1818*a1^2*b2*c*n^2-90*a1^3*b2*c*n^2-18045*a2\
*b2*c*n^2+7584*a1*a2*b2*c*n^2-783*a1^2*a2*b2*c*n^2+16*a1^3*a2*b2*c*n^2+6640*a2^2\
*b2*c*n^2-1575*a1*a2^2*b2*c*n^2+62*a1^2*a2^2*b2*c*n^2-900*a2^3*b2*c*n^2+76*a1*a2\
^3*b2*c*n^2+30*a2^4*b2*c*n^2-7776*b1*b2*c*n^2+3948*a1*b1*b2*c*n^2-513*a1^2*b1*b2\
*c*n^2+16*a1^3*b1*b2*c*n^2+7584*a2*b1*b2*c*n^2-2304*a1*a2*b1*b2*c*n^2+124*a1^2*a\
2*b1*b2*c*n^2-2295*a2^2*b1*b2*c*n^2+296*a1*a2^2*b1*b2*c*n^2+204*a2^3*b1*b2*c*n^2\
+874*b1^2*b2*c*n^2-279*a1*b1^2*b2*c*n^2+14*a1^2*b1^2*b2*c*n^2-783*a2*b1^2*b2*c*n\
^2+124*a1*a2*b1^2*b2*c*n^2+160*a2^2*b1^2*b2*c*n^2-18*b1^3*b2*c*n^2+16*a2*b1^3*b2\
*c*n^2-7776*b2^2*c*n^2+4414*a1*b2^2*c*n^2-693*a1^2*b2^2*c*n^2+25*a1^3*b2^2*c*n^2\
+6640*a2*b2^2*c*n^2-2295*a1*a2*b2^2*c*n^2+160*a1^2*a2*b2^2*c*n^2-1710*a2^2*b2^2*\
c*n^2+259*a1*a2^2*b2^2*c*n^2+124*a2^3*b2^2*c*n^2+2226*b1*b2^2*c*n^2-909*a1*b1*b2\
^2*c*n^2+85*a1^2*b1*b2^2*c*n^2-1575*a2*b1*b2^2*c*n^2+296*a1*a2*b1*b2^2*c*n^2+259\
*a2^2*b1*b2^2*c*n^2-108*b1^2*b2^2*c*n^2+14*a1*b1^2*b2^2*c*n^2+62*a2*b1^2*b2^2*c*\
n^2+1484*b2^3*c*n^2-720*a1*b2^3*c*n^2+80*a1^2*b2^3*c*n^2-900*a2*b2^3*c*n^2+204*a\
1*a2*b2^3*c*n^2+124*a2^2*b2^3*c*n^2-180*b1*b2^3*c*n^2+44*a1*b1*b2^3*c*n^2+76*a2*\
b1*b2^3*c*n^2-90*b2^4*c*n^2+30*a1*b2^4*c*n^2+30*a2*b2^4*c*n^2+2994*c^2*n^2-1485*\
a1*c^2*n^2+253*a1^2*c^2*n^2-18*a1^3*c^2*n^2-2970*a2*c^2*n^2+874*a1*a2*c^2*n^2-72\
*a1^2*a2*c^2*n^2+2*a1^3*a2*c^2*n^2+874*a2^2*c^2*n^2-108*a1*a2^2*c^2*n^2+2*a1^2*a\
2^2*c^2*n^2-72*a2^3*c^2*n^2-1485*b1*c^2*n^2+368*a1*b1*c^2*n^2-18*a1^2*b1*c^2*n^2\
+2*a1^3*b1*c^2*n^2+1510*a2*b1*c^2*n^2-189*a1*a2*b1*c^2*n^2-3*a1^2*a2*b1*c^2*n^2-\
405*a2^2*b1*c^2*n^2+10*a1*a2^2*b1*c^2*n^2+24*a2^3*b1*c^2*n^2+253*b1^2*c^2*n^2-18\
*a1*b1^2*c^2*n^2-4*a1^2*b1^2*c^2*n^2-252*a2*b1^2*c^2*n^2+14*a1*a2*b1^2*c^2*n^2+4\
9*a2^2*b1^2*c^2*n^2-18*b1^3*c^2*n^2+2*a1*b1^3*c^2*n^2+11*a2*b1^3*c^2*n^2-2970*b2\
*c^2*n^2+1510*a1*b2*c^2*n^2-252*a1^2*b2*c^2*n^2+11*a1^3*b2*c^2*n^2+2384*a2*b2*c^\
2*n^2-693*a1*a2*b2*c^2*n^2+47*a1^2*a2*b2*c^2*n^2-513*a2^2*b2*c^2*n^2+58*a1*a2^2*\
b2*c^2*n^2+24*a2^3*b2*c^2*n^2+874*b1*b2*c^2*n^2-189*a1*b1*b2*c^2*n^2+14*a1^2*b1*\
b2*c^2*n^2-693*a2*b1*b2*c^2*n^2+72*a1*a2*b1*b2*c^2*n^2+108*a2^2*b1*b2*c^2*n^2-72\
*b1^2*b2*c^2*n^2-3*a1*b1^2*b2*c^2*n^2+47*a2*b1^2*b2*c^2*n^2+2*b1^3*b2*c^2*n^2+87\
4*b2^2*c^2*n^2-405*a1*b2^2*c^2*n^2+49*a1^2*b2^2*c^2*n^2-513*a2*b2^2*c^2*n^2+108*\
a1*a2*b2^2*c^2*n^2+61*a2^2*b2^2*c^2*n^2-108*b1*b2^2*c^2*n^2+10*a1*b1*b2^2*c^2*n^\
2+58*a2*b1*b2^2*c^2*n^2+2*b1^2*b2^2*c^2*n^2-72*b2^3*c^2*n^2+24*a1*b2^3*c^2*n^2+2\
4*a2*b2^3*c^2*n^2-189*c^3*n^2+66*a1*c^3*n^2-18*a1^2*c^3*n^2+a1^3*c^3*n^2+132*a2*\
c^3*n^2-18*a1*a2*c^3*n^2+2*a1^2*a2*c^3*n^2-18*a2^2*c^3*n^2+66*b1*c^3*n^2+18*a1*b\
1*c^3*n^2-a1^2*b1*c^3*n^2-54*a2*b1*c^3*n^2-4*a1*a2*b1*c^3*n^2+6*a2^2*b1*c^3*n^2-\
18*b1^2*c^3*n^2-a1*b1^2*c^3*n^2+8*a2*b1^2*c^3*n^2+b1^3*c^3*n^2+132*b2*c^3*n^2-54\
*a1*b2*c^3*n^2+8*a1^2*b2*c^3*n^2-72*a2*b2*c^3*n^2+12*a1*a2*b2*c^3*n^2+6*a2^2*b2*\
c^3*n^2-18*b1*b2*c^3*n^2-4*a1*b1*b2*c^3*n^2+12*a2*b1*b2*c^3*n^2+2*b1^2*b2*c^3*n^\
2-18*b2^2*c^3*n^2+6*a1*b2^2*c^3*n^2+6*a2*b2^2*c^3*n^2+20745*n^3-14520*a1*n^3+278\
1*a1^2*n^3-132*a1^3*n^3-29040*a2*n^3+15234*a1*a2*n^3-2040*a1^2*a2*n^3+60*a1^3*a2\
*n^3+15234*a2^2*n^3-5328*a1*a2^2*n^3+396*a1^2*a2^2*n^3-4*a1^3*a2^2*n^3-3552*a2^3\
*n^3+672*a1*a2^3*n^3-16*a1^2*a2^3*n^3+336*a2^4*n^3-20*a1*a2^4*n^3-8*a2^5*n^3-145\
20*b1*n^3+9672*a1*b1*n^3-1644*a1^2*b1*n^3+60*a1^3*b1*n^3+17289*a2*b1*n^3-8240*a1\
*a2*b1*n^3+894*a1^2*a2*b1*n^3-16*a1^3*a2*b1*n^3-7616*a2^2*b1*n^3+2196*a1*a2^2*b1\
*n^3-102*a1^2*a2^2*b1*n^3+1392*a2^3*b1*n^3-168*a1*a2^3*b1*n^3-82*a2^4*b1*n^3+278\
1*b1^2*n^3-1644*a1*b1^2*n^3+216*a1^2*b1^2*n^3-4*a1^3*b1^2*n^3-2664*a2*b1^2*n^3+1\
020*a1*a2*b1^2*n^3-66*a1^2*a2*b1^2*n^3+918*a2^2*b1^2*n^3-164*a1*a2^2*b1^2*n^3-10\
6*a2^3*b1^2*n^3-132*b1^3*n^3+60*a1*b1^3*n^3-4*a1^2*b1^3*n^3+90*a2*b1^3*n^3-20*a1\
*a2*b1^3*n^3-20*a2^2*b1^3*n^3-29040*b2*n^3+17289*a1*b2*n^3-2664*a1^2*b2*n^3+90*a\
1^3*b2*n^3+32523*a2*b2*n^3-13568*a1*a2*b2*n^3+1290*a1^2*a2*b2*n^3-20*a1^3*a2*b2*\
n^3-12944*a2^2*b2*n^3+3204*a1*a2^2*b2*n^3-126*a1^2*a2^2*b2*n^3+2064*a2^3*b2*n^3-\
208*a1*a2^3*b2*n^3-102*a2^4*b2*n^3+15234*b1*b2*n^3-8240*a1*b1*b2*n^3+1020*a1^2*b\
1*b2*n^3-20*a1^3*b1*b2*n^3-13568*a2*b1*b2*n^3+4656*a1*a2*b1*b2*n^3-274*a1^2*a2*b\
1*b2*n^3+4032*a2^2*b1*b2*n^3-630*a1*a2^2*b1*b2*n^3-380*a2^3*b1*b2*n^3-2040*b1^2*\
b2*n^3+894*a1*b1^2*b2*n^3-66*a1^2*b1^2*b2*n^3+1290*a2*b1^2*b2*n^3-274*a1*a2*b1^2\
*b2*n^3-224*a2^2*b1^2*b2*n^3+60*b1^3*b2*n^3-16*a1*b1^3*b2*n^3-20*a2*b1^3*b2*n^3+\
15234*b2^2*n^3-7616*a1*b2^2*n^3+918*a1^2*b2^2*n^3-20*a1^3*b2^2*n^3-12944*a2*b2^2\
*n^3+4032*a1*a2*b2^2*n^3-224*a1^2*a2*b2^2*n^3+3510*a2^2*b2^2*n^3-490*a1*a2^2*b2^\
2*n^3-290*a2^3*b2^2*n^3-5328*b1*b2^2*n^3+2196*a1*b1*b2^2*n^3-164*a1^2*b1*b2^2*n^\
3+3204*a2*b1*b2^2*n^3-630*a1*a2*b1*b2^2*n^3-490*a2^2*b1*b2^2*n^3+396*b1^2*b2^2*n\
^3-102*a1*b1^2*b2^2*n^3-126*a2*b1^2*b2^2*n^3-4*b1^3*b2^2*n^3-3552*b2^3*n^3+1392*\
a1*b2^3*n^3-106*a1^2*b2^3*n^3+2064*a2*b2^3*n^3-380*a1*a2*b2^3*n^3-290*a2^2*b2^3*\
n^3+672*b1*b2^3*n^3-168*a1*b1*b2^3*n^3-208*a2*b1*b2^3*n^3-16*b1^2*b2^3*n^3+336*b\
2^4*n^3-82*a1*b2^4*n^3-102*a2*b2^4*n^3-20*b1*b2^4*n^3-8*b2^5*n^3-14520*c*n^3+761\
7*a1*c*n^3-1020*a1^2*c*n^3+30*a1^3*c*n^3+15234*a2*c*n^3-5328*a1*a2*c*n^3+396*a1^\
2*a2*c*n^3-4*a1^3*a2*c*n^3-5328*a2^2*c*n^3+1008*a1*a2^2*c*n^3-24*a1^2*a2^2*c*n^3\
+672*a2^3*c*n^3-40*a1*a2^3*c*n^3-20*a2^4*c*n^3+7617*b1*c*n^3-3288*a1*b1*c*n^3+30\
6*a1^2*b1*c*n^3-4*a1^3*b1*c*n^3-6992*a2*b1*c*n^3+1776*a1*a2*b1*c*n^3-62*a1^2*a2*\
b1*c*n^3+1980*a2^2*b1*c*n^3-202*a1*a2^2*b1*c*n^3-160*a2^3*b1*c*n^3-1020*b1^2*c*n\
^3+306*a1*b1^2*c*n^3-12*a1^2*b1^2*c*n^3+816*a2*b1^2*c*n^3-114*a1*a2*b1^2*c*n^3-1\
54*a2^2*b1^2*c*n^3+30*b1^3*c*n^3-4*a1*b1^3*c*n^3-20*a2*b1^3*c*n^3+15234*b2*c*n^3\
-6992*a1*b2*c*n^3+816*a1^2*b2*c*n^3-20*a1^3*b2*c*n^3-12320*a2*b2*c*n^3+3408*a1*a\
2*b2*c*n^3-174*a1^2*a2*b2*c*n^3+2988*a2^2*b2*c*n^3-350*a1*a2^2*b2*c*n^3-200*a2^3\
*b2*c*n^3-5328*b1*b2*c*n^3+1776*a1*b1*b2*c*n^3-114*a1^2*b1*b2*c*n^3+3408*a2*b1*b\
2*c*n^3-512*a1*a2*b1*b2*c*n^3-510*a2^2*b1*b2*c*n^3+396*b1^2*b2*c*n^3-62*a1*b1^2*\
b2*c*n^3-174*a2*b1^2*b2*c*n^3-4*b1^3*b2*c*n^3-5328*b2^2*c*n^3+1980*a1*b2^2*c*n^3\
-154*a1^2*b2^2*c*n^3+2988*a2*b2^2*c*n^3-510*a1*a2*b2^2*c*n^3-380*a2^2*b2^2*c*n^3\
+1008*b1*b2^2*c*n^3-202*a1*b1*b2^2*c*n^3-350*a2*b1*b2^2*c*n^3-24*b1^2*b2^2*c*n^3\
+672*b2^3*c*n^3-160*a1*b2^3*c*n^3-200*a2*b2^3*c*n^3-40*b1*b2^3*c*n^3-20*b2^4*c*n\
^3+2781*c^2*n^3-1020*a1*c^2*n^3+114*a1^2*c^2*n^3-4*a1^3*c^2*n^3-2040*a2*c^2*n^3+\
396*a1*a2*c^2*n^3-16*a1^2*a2*c^2*n^3+396*a2^2*c^2*n^3-24*a1*a2^2*c^2*n^3-16*a2^3\
*c^2*n^3-1020*b1*c^2*n^3+168*a1*b1*c^2*n^3-4*a1^2*b1*c^2*n^3+678*a2*b1*c^2*n^3-4\
2*a1*a2*b1*c^2*n^3-90*a2^2*b1*c^2*n^3+114*b1^2*c^2*n^3-4*a1*b1^2*c^2*n^3-56*a2*b\
1^2*c^2*n^3-4*b1^3*c^2*n^3-2040*b2*c^2*n^3+678*a1*b2*c^2*n^3-56*a1^2*b2*c^2*n^3+\
1074*a2*b2*c^2*n^3-154*a1*a2*b2*c^2*n^3-114*a2^2*b2*c^2*n^3+396*b1*b2*c^2*n^3-42\
*a1*b1*b2*c^2*n^3-154*a2*b1*b2*c^2*n^3-16*b1^2*b2*c^2*n^3+396*b2^2*c^2*n^3-90*a1\
*b2^2*c^2*n^3-114*a2*b2^2*c^2*n^3-24*b1*b2^2*c^2*n^3-16*b2^3*c^2*n^3-132*c^3*n^3\
+30*a1*c^3*n^3-4*a1^2*c^3*n^3+60*a2*c^3*n^3-4*a1*a2*c^3*n^3-4*a2^2*c^3*n^3+30*b1\
*c^3*n^3+4*a1*b1*c^3*n^3-12*a2*b1*c^3*n^3-4*b1^2*c^3*n^3+60*b2*c^3*n^3-12*a1*b2*\
c^3*n^3-16*a2*b2*c^3*n^3-4*b1*b2*c^3*n^3-4*b2^2*c^3*n^3+18150*n^4-10035*a1*n^4+1\
431*a1^2*n^4-45*a1^3*n^4-20070*a2*n^4+7804*a1*a2*n^4-690*a1^2*a2*n^4+10*a1^3*a2*\
n^4+7804*a2^2*n^4-1800*a1*a2^2*n^4+66*a1^2*a2^2*n^4-1200*a2^3*n^4+112*a1*a2^3*n^\
4+56*a2^4*n^4-10035*b1*n^4+4942*a1*b1*n^4-555*a1^2*b1*n^4+10*a1^3*b1*n^4+8844*a2\
*b1*n^4-2775*a1*a2*b1*n^4+149*a1^2*a2*b1*n^4-2565*a2^2*b1*n^4+366*a1*a2^2*b1*n^4\
+232*a2^3*b1*n^4+1431*b1^2*n^4-555*a1*b1^2*n^4+36*a1^2*b1^2*n^4-900*a2*b1^2*n^4+\
170*a1*a2*b1^2*n^4+153*a2^2*b1^2*n^4-45*b1^3*n^4+10*a1*b1^3*n^4+15*a2*b1^3*n^4-2\
0070*b2*n^4+8844*a1*b2*n^4-900*a1^2*b2*n^4+15*a1^3*b2*n^4+16648*a2*b2*n^4-4575*a\
1*a2*b2*n^4+215*a1^2*a2*b2*n^4-4365*a2^2*b2*n^4+534*a1*a2^2*b2*n^4+344*a2^3*b2*n\
^4+7804*b1*b2*n^4-2775*a1*b1*b2*n^4+170*a1^2*b1*b2*n^4-4575*a2*b1*b2*n^4+776*a1*\
a2*b1*b2*n^4+672*a2^2*b1*b2*n^4-690*b1^2*b2*n^4+149*a1*b1^2*b2*n^4+215*a2*b1^2*b\
2*n^4+10*b1^3*b2*n^4+7804*b2^2*n^4-2565*a1*b2^2*n^4+153*a1^2*b2^2*n^4-4365*a2*b2\
^2*n^4+672*a1*a2*b2^2*n^4+585*a2^2*b2^2*n^4-1800*b1*b2^2*n^4+366*a1*b1*b2^2*n^4+\
534*a2*b1*b2^2*n^4+66*b1^2*b2^2*n^4-1200*b2^3*n^4+232*a1*b2^3*n^4+344*a2*b2^3*n^\
4+112*b1*b2^3*n^4+56*b2^4*n^4-10035*c*n^4+3902*a1*c*n^4-345*a1^2*c*n^4+5*a1^3*c*\
n^4+7804*a2*c*n^4-1800*a1*a2*c*n^4+66*a1^2*a2*c*n^4-1800*a2^2*c*n^4+168*a1*a2^2*\
c*n^4+112*a2^3*c*n^4+3902*b1*c*n^4-1110*a1*b1*c*n^4+51*a1^2*b1*c*n^4-2355*a2*b1*\
c*n^4+296*a1*a2*b1*c*n^4+330*a2^2*b1*c*n^4-345*b1^2*c*n^4+51*a1*b1^2*c*n^4+136*a\
2*b1^2*c*n^4+5*b1^3*c*n^4+7804*b2*c*n^4-2355*a1*b2*c*n^4+136*a1^2*b2*c*n^4-4155*\
a2*b2*c*n^4+568*a1*a2*b2*c*n^4+498*a2^2*b2*c*n^4-1800*b1*b2*c*n^4+296*a1*b1*b2*c\
*n^4+568*a2*b1*b2*c*n^4+66*b1^2*b2*c*n^4-1800*b2^2*c*n^4+330*a1*b2^2*c*n^4+498*a\
2*b2^2*c*n^4+168*b1*b2^2*c*n^4+112*b2^3*c*n^4+1431*c^2*n^4-345*a1*c^2*n^4+19*a1^\
2*c^2*n^4-690*a2*c^2*n^4+66*a1*a2*c^2*n^4+66*a2^2*c^2*n^4-345*b1*c^2*n^4+28*a1*b\
1*c^2*n^4+113*a2*b1*c^2*n^4+19*b1^2*c^2*n^4-690*b2*c^2*n^4+113*a1*b2*c^2*n^4+179\
*a2*b2*c^2*n^4+66*b1*b2*c^2*n^4+66*b2^2*c^2*n^4-45*c^3*n^4+5*a1*c^3*n^4+10*a2*c^\
3*n^4+5*b1*c^3*n^4+10*b2*c^3*n^4+10035*n^5-4110*a1*n^5+387*a1^2*n^5-6*a1^3*n^5-8\
220*a2*n^5+2106*a1*a2*n^5-92*a1^2*a2*n^5+2106*a2^2*n^5-240*a1*a2^2*n^5-160*a2^3*\
n^5-4110*b1*n^5+1332*a1*b1*n^5-74*a1^2*b1*n^5+2385*a2*b1*n^5-370*a1*a2*b1*n^5-34\
2*a2^2*b1*n^5+387*b1^2*n^5-74*a1*b1^2*n^5-120*a2*b1^2*n^5-6*b1^3*n^5-8220*b2*n^5\
+2385*a1*b2*n^5-120*a1^2*b2*n^5+4491*a2*b2*n^5-610*a1*a2*b2*n^5-582*a2^2*b2*n^5+\
2106*b1*b2*n^5-370*a1*b1*b2*n^5-610*a2*b1*b2*n^5-92*b1^2*b2*n^5+2106*b2^2*n^5-34\
2*a1*b2^2*n^5-582*a2*b2^2*n^5-240*b1*b2^2*n^5-160*b2^3*n^5-4110*c*n^5+1053*a1*c*\
n^5-46*a1^2*c*n^5+2106*a2*c*n^5-240*a1*a2*c*n^5-240*a2^2*c*n^5+1053*b1*c*n^5-148\
*a1*b1*c*n^5-314*a2*b1*c*n^5-46*b1^2*c*n^5+2106*b2*c*n^5-314*a1*b2*c*n^5-554*a2*\
b2*c*n^5-240*b1*b2*c*n^5-240*b2^2*c*n^5+387*c^2*n^5-46*a1*c^2*n^5-92*a2*c^2*n^5-\
46*b1*c^2*n^5-92*b2*c^2*n^5-6*c^3*n^5+3425*n^6-924*a1*n^6+43*a1^2*n^6-1848*a2*n^\
6+234*a1*a2*n^6+234*a2^2*n^6-924*b1*n^6+148*a1*b1*n^6+265*a2*b1*n^6+43*b1^2*n^6-\
1848*b2*n^6+265*a1*b2*n^6+499*a2*b2*n^6+234*b1*b2*n^6+234*b2^2*n^6-924*c*n^6+117\
*a1*c*n^6+234*a2*c*n^6+117*b1*c*n^6+234*b2*c*n^6+43*c^2*n^6+660*n^7-88*a1*n^7-17\
6*a2*n^7-88*b1*n^7-176*b2*n^7-88*c*n^7+55*n^8)*N-((-1+a1-n)*(-1+a2-n)*(-1+b1-n)\
*(-1+b2-n)*(-1+a2+b2+c-n)*(-40+12*a1+24*a2-4*a1*a2-4*a2^2+12*b1-4*a1*b1-6*a2*b1+\
a1*a2*b1+a2^2*b1+24*b2-6*a1*b2-10*a2*b2+a1*a2*b2+a2^2*b2-4*b1*b2+a1*b1*b2+a2*b1*\
b2-4*b2^2+a1*b2^2+a2*b2^2+12*c-2*a1*c-4*a2*c-2*b1*c+a2*b1*c-4*b2*c+a1*b2*c+a2*b2\
*c-60*n+12*a1*n+24*a2*n-2*a1*a2*n-2*a2^2*n+12*b1*n-2*a1*b1*n-3*a2*b1*n+24*b2*n-3\
*a1*b2*n-5*a2*b2*n-2*b1*b2*n-2*b2^2*n+12*c*n-a1*c*n-2*a2*c*n-b1*c*n-2*b2*c*n-30*\
n^2+3*a1*n^2+6*a2*n^2+3*b1*n^2+6*b2*n^2+3*c*n^2-5*n^3)):
NorOp(ope,N):
end proc:

                     

###START GENERAL THEOREM

#TheoremZ2g(a1,a2,b1,b2,e,K): Inputs rational numbers a1,a2,b1,b2,c, and a large positive integer K and outputs
#a theorem regarding the constant IntGBg(a1,a2,b1,b2,e,0) (q.v.)`):
#Either a suggested proof of irrationality or a way to compute it exponentially  fast.
#Try:
#TheoremZ2g(0,0,0,0,0,2000):
TheoremZ2g:=proc(a1,a2,b1,b2,e,K) local n,N,gu,x,y,ope,lu,X,A,B,beta,F,c,E,A1,B1,delta,d1,ka:
beta:=BC():
gu:=CnDnG(a1,a2,b1,b2,e,K):
lu:=AnBnG(a1,a2,b1,b2,e,K):
ka:=GBCg(a1,a2,b1,b2,e):


 ope:=OPEZ2g(a1,a2,b1,b2,e,n,N):

print(``):
print(`-------------------------------------------------------------------------------`):
print(``):
if min(op(gu[3]))<0 then
 print(`Very fast Computation of rational approximations to the constant `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1),x=0..1),y=0..1)):
print(`divided by `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
 print(``):
 print(`By Shalosh B. Ekhad `):
 print(``):


 print(`Theorem: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates  the constant`):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1),x=0..1),y=0..1)):
print(`divided by `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):


if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

print(`According to Maple, this constant equals `):
print(``):
print(IntGBg(a1,a2,b1,b2,e,0)):
print(``):
print(`and in Maple format `):
print(``):
lprint(IntGBg(a1,a2,b1,b2,e,0)):
print(``):

print(`Comment: while this sequence does not lead to an irrationality proof, for the record, the delta, happens to be roughly`):
print(``):
print(min(op(gu[3]))):
print(``):

 print(`Proof, consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(`normalized by dividing by the following constant, (independent of n) `):
 print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):

 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):

else


 print(`Sketch of an Irrationality Proof of the constant `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1),x=0..1),y=0..1)):
print(`divided by `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):

  print(``):
 print(`By Shalosh B. Ekhad `):
   print(``):

 print(`Theorem: The constant of the title `):
 print(`is irrational, with an irrationality measure`, 1+ (log(beta)+nu)/(log(beta)-nu), `for a certain number nu `):
print(`that is approximately `, max(op(gu[2])) , ` yielding an irrationality measure that is approximately `, evalf(1+1/min(op(gu[3])),10)):
print(``):
 print(`We hope that the reader can find  nu exactly. `):

if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:


print(`According to Maple, this constant equals `):

print(``):
print(IntGBg(a1,a2,b1,b2,e,0)):
print(``):
print(`and in Maple format `):
print(``):
lprint(IntGBg(a1,a2,b1,b2,e,0)):
print(``):

 print(`We need two  lemmas `):
    print(``):

 print(`Lemma: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates the constant of the title, c`):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):
 print(`Proof, consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(`normalized by dividing by the following constant, (independent of n) `):
 print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):
print(``):
 print(`we now claim that the sequence of RATIONAL numbers A(n),B(n), can be multiplied by another sequence of rational numbers `):
 print(`E(n) such that both A(n)E(n) and B(n)E(n) are integers `):
 print(``):
 print(`Lemma: There exists a sequence of rational numbers, whose prime factorizations consists of small primes, that hopefully`):
 print(`can be described (and proved) explicity, that we leave to the expert reader such that `):
 print(` A1(n):=E(n)A(n), B1(n):=E(n)B(n) are BOTH integers`):
 print(``):
 print(`Furthermore there exists a contant, nu, that hopefully the learned reader can determine such that E(n) is OMEGA of `, exp(nu*n) ):
 print(``):
 print(`The empircal values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n) we get `):
print(``):
print( E(n)*F(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nu)/(log(beta) + nu) ):
print(``):
print(`Using the above values of nu for E(n) from`, K+1-nops(gu[3]), `to `, K+1, `the estimated deltas are `):
print(``):
print(gu[3]):
print(``):
print(`As you can see, they are all positive `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):
fi:

end:



#TheoremZ2gnum(a1,a2,b1,b2,e,K,num): Inputs rational numbers a1,a2,b1,b2,c, and a large positive integer K and outputs
#a numbered theorem regarding the constant IntGBg(a1,a2,b1,b2,e,0) (q.v.)`):
#Either a suggested proof of irrationality or a way to compute it exponentially  fast.
#Try:
#TheoremZ2gnum(0,0,0,0,0,2000,num):
TheoremZ2gnum:=proc(a1,a2,b1,b2,e,K,num) local n,N,gu,x,y,ope,lu,X,A,B,beta,F,c,E,A1,B1,delta,d1,ka:
beta:=BC():
gu:=CnDnG(a1,a2,b1,b2,e,K):
lu:=AnBnG(a1,a2,b1,b2,e,K):
ka:=GBCg(a1,a2,b1,b2,e):


 ope:=OPEZ2g(a1,a2,b1,b2,e,n,N):

if min(op(gu[3]))<0 then

 print(`Theorem Number`, num , `:  let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates  the constant`):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1),x=0..1),y=0..1)):
print(`divided by `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):


if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

 print(`Proof, consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(`normalized by dividing by the following constant, (independent of n) `):
 print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):

 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):

else


 print(`Theorem number`, num, `: The following constant c. `):
   print(``):
 print( c=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1),x=0..1),y=0..1)  ):
print(``):
print(`divided by `):
 print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
    print(``):
 print(`is irrational, with an irrationality measure`, 1+ (log(beta)+nu)/(log(beta)-nu), `for a certain number nu `):
print(`that is approximately `, max(op(gu[2])) , ` yielding an irrationality measure that is approximately `, evalf(1+1/min(op(gu[3])),10)):

print(``):
 print(`We hope that the reader can find  nu exactly.`):

 print(`We hope that the reader can find  nu exactly. `):

if identify(ka)<>ka then
  print(`Comment: Note that this constant appears to be `, identify(ka) ):
  print(`Prove it!`):
fi:

 print(`We need two  lemmas `):
    print(``):

 print(`Lemma: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`):
 print(``):
 print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0):
 print(``):
 print(`Subject to the initial conditions`):
  print(``):
 print(A(0)=lu[1][1],A(1)=lu[1][2]):
  print(``):
  print(B(0)=lu[2][1],B(1)=lu[2][2]):
 print(``):
 print(`Then`, A(n)/B(n), `approximates the constant of the title, c`):
 print(``):
  print(`with an error that is OMEGA of`, (1/BC()^2)^n, ` that in floating point is`,  evalf((1/BC()^2)^n) ):
 print(``):
 print(`Proof, consider the Beukers type-integral `):
 print(F(n)=Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^(e+1)*(x*(1-x)*y*(1-y)/(1-x*y))^n,x=0..1),y=0..1)):
print(`normalized by dividing by the following constant, (independent of n) `):
 print(Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y)^e,x=0..1),y=0..1)):
 print(``):
 print(`Then `, F(0)=B(0)*c-A(0), F(1)=c*B(1)-A(1) ):
 print(``):
 print(`and F(n) also satisfies the above recurrence, thanks to the amazing multivariable Almkvist-Zeilberger algorithm `):
 print(``):
  print(`Hence`, F(n)=B(n)*c-A(n)):
 print(``):
  print(`By a simple bound of the integrand,  F(n) is OMEGA of`, 1/beta^n, ` and  by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n):
 print(``):
 print(`Dividing by B(n) gives that A(n)/B(n)-c is OMEGA of `, 1/beta^(2*n) , `QED. `):
print(``):
 print(`we now claim that the sequence of RATIONAL numbers A(n),B(n), can be multiplied by another sequence of rational numbers `):
 print(`E(n) such that both A(n)E(n) and B(n)E(n) are integers `):
 print(``):
 print(`Lemma: There exists a sequence of rational numbers, whose prime factorizations consists of small primes, that hopefully`):
 print(`can be described (and proved) explicity, that we leave to the expert reader such that `):
 print(` A1(n):=E(n)A(n), B1(n):=E(n)B(n) are BOTH integers`):
 print(``):
 print(`Furthermore there exists a contant, nu, that hopefully the learned reader can determine such that E(n) is OMEGA of `, exp(nu*n) ):
 print(``):
 print(`The empircal values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `):
 print(``):
 print(gu[2]):
print(``):
 print(`Multiplying F(n) by E(n) we get `):
print(``):
print( E(n)*F(n)=B1(n)*c-A1(n)):
print(``):
print(`and this implies that `):
print(``):
print(abs(c-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)):
print(``):
print(` where `, delta= (log(beta)-nu)/(log(beta) + nu) ):
print(``):
print(`Using the above values of nu for E(n) from`, K+1-nops(gu[3]), `to `, K+1, `the estimated deltas are `):
print(``):
print(gu[3]):
print(``):
print(`As you can see, they are all positive `):
print(``):
print(`We leave it to the reader to fill-in the details.`):
print(``):
fi:

end:



#PaperZ2g(L,K): Given a list L of pentuples that are believed to be provably irrational and a large positive ineger K (around 2000 is OK)
# outputs a paper with sketches
#of proof. Try:
#PaperZ2g([[0,0,0,0,0],[0,0,0,0,1/2]],2000):
PaperZ2g:=proc(L,K) local i,t0:
t0:=time():
print(``):
print(`Sketches of proofs of the Irrationality of `, nops(L), `Constants given as certain double integrals`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for i from 1 to nops(L) do
print(``):
print(`------------------------------------------------------------`):
print(``):
TheoremZ2gnum(op(L[i]),K,i):
print(``):
od:

print(``):
print(`-----------------------------------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to generate `):
print(``):
end:









#Search(N,K): All the 4-tuples [a,b,c,d]=[a1/N,a2/N,a3/N,a4/N] with the ai's beteen -(N-1) and N-1 such that
#the deltas for AnBn(a,b,c,d,e) seems to be positive. Try
#Search(2,100);

Search:=proc(N,K) local ALD, i,mu,a1,a2,a3,a4,gu1,gu:

ALD:={}:
gu:=[]:
for a1 from -(N-1) to N-1 do
for a2 from -(N-1) to N-1 do
for a3 from -(N-1) to N-1 do
for a4 from -(N-1) to N-1 do

gu1:=[a1/N,a2/N,a3/N,a4/N]:


if not member(gu1,ALD) then

if lcm(seq(denom(gu1[i]),i=1..nops(gu1)))=N then

mu:=CnDn(op(gu1),K):
if mu<>FAIL then

if min(op(mu[3]))>0 then
  gu:=[op(gu), gu1]:
ALD:=ALD union {gu1, [gu1[3],gu1[4],gu1[1],gu1[2]]}:

fi:
fi:
fi:
fi:

od:
od:
od:
od:

gu:
end:





#MyIDs(C,F,F0,N): Given a constant C in decimals and another constant
#let's call it F, whose floating-point if F0
# and a positive integer N
#tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ
#MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);
MyIDs:=proc(C,F,F0,N) local gu,mu,i:

gu:=IntegerRelations[PSLQ]([1,evalf(C),evalf(F0),evalf(C*F0)]):


if max(seq(abs(gu[i]),i=1..4)) >N then
 RETURN(FAIL):
fi:

mu:=-(gu[1]+gu[3]*F)/(gu[2]+gu[4]*F):

if abs(evalf(subs(F=F0,mu)-C))>1/10^(Digits-7) then
  RETURN(FAIL):
fi:

mu:

end:


#SortC(L,N,r): Given a list of 4-tuples, L, and a positive integer N, and a symbol r
#divides them into classes such that GBC(op(L[i])) are related to
#each other (conjecturally) by a fractional-linear transfomation with integers  less than N in absolute value
#The output is a list of lists such that the first entry is the main pentuple, followed by the symbol r[i],
#its decimal approximation, followed by the tuples, and the expression in terms of r[i]
#Try:
#SortC(Hopefuls5()[2],10000,r);
SortC:=proc(L,N,r) local i,S1,Rf,lu1,lu2,co,T,ka:


S1:=convert(L,set):

co:=0:
while S1<>{} do
co:=co+1:
 lu1:=S1[1]:
 Rf:=GBC(op(lu1)):
 T[co]:=[]:

 for lu2 in S1   do
   ka:=MyIDs(GBC(op(lu2)),r[co],Rf,N):
  if ka<>FAIL then
   T[co]:=[op(T[co]),[lu2,ka]]:
  S1:=S1 minus {lu2}:
 fi:
od:

od:

[seq(T[i],i=1..co)]:

end:



###START PRIMES PROCEDURES


#PpG(a,b,n,m1,m2,C,M): Given rational numbers a and b between 0 and 1 and positive integers m1,m2 and n
#and a subset C in {1,...M-1} and an integer M
#it is the product of p between m1 and m2  such that frac{n/p} is between a and b
#and p mod M is in C
PpG:=proc(a,b,n,m1,m2,C,M) local p,gu:
p:=2:

gu:=1:

p:=nextprime(m1):


while p<m2 do
 if (frac(n/p)>= a and frac(n/p)<b and member(p mod M, C)) then
 gu:=gu*p:
fi:
p:=nextprime(p):
od:

gu:

end:

#Lc(n): the lcm of 1, ..., n. Try: Lc(1000);
Lc:=proc(n) local i:
lcm(seq(i,i=1..n)):
end:


#LcAB(n,A,B): the lcm of seq(A+B*i,i=1..n). Try: LcAB(1000,1,2);
LcAB:=proc(n,A,B) local i:
lcm(seq(A+B*i,i=1..n)):
end:


#PrimesF(N,n,k): inputs a HUGE integer N that is a product of small primes, and a much smaller integer n
#that generated N (via some process), outputs the list of length k, whose i-th entry is the list
#of primes>=sqrt(n) that show up with exponent i, followed by the list of primes that
#did not show up at all. Try:
#lu:=op(Hopefuls2()[3][1]): N:=CnDn(op(lu),2000)[1][2][-1],2000): PrimesF(N,2000,6);

PrimesF:=proc(N,n,k) local gu,PR,KULAM,T,i,gadol,katan,j,p:

gu:=ifactors(N)[2]:

if nops(gu)=1 then
 RETURN(gu):
fi:

PR:={}:

for i from 1 to k do
T[i]:=[]:
od:

for j from 1 to nops(gu) do
 if gu[j][1]>=evalf(sqrt(n)) then
    T[gu[j][2]]:=[op(T[gu[j][2]]),gu[j][1]]:
  PR:=PR union {gu[j][1]}:
 fi:
od:

gadol:=max(op(PR)):
katan:=min(op(PR)):

KULAM:={}:

p:=nextprime(katan):
KULAM:={p}:

while p<=gadol do
 p:=nextprime(p):
 KULAM:=KULAM union {p}:
od:

PR:=KULAM minus PR:

[ [seq(T[i],i=1..k)],PR]:

end:



#Fs(S,n): Given a set S and a positive integer n finds sort([seq(evalf(frac(n/p),10),p in S)]);
Fs:=proc(S,n) local p:
sort([seq(evalf(frac(n/p),10),p in S)]);
end:


#Seg1(L,a,b): Given a  list L and integers a<b, extracts the segment between a and b. Try:
#Seg1([seq(i,i=1..100)],40,60);
Seg1:=proc(L,a,b) local L1,i,j:
L1:=sort(L):

if a>b then
 RETURN(FAIL):
fi:

for i from 1 while L1[i]<a do od:

for j from i while L[j]<b do od:

[op(i..j-1,L)]:
end:




#EvalPpG1(P,n1): evaluates a partial  PpG object at n=n1. 
#For GBC(0,0,0,0) (alias Zeta(2))
#EvalPpG1([[1, 1], [1, 2],[] ],1000  );

EvalPpG1:=proc(P,n1) local gu,lu,i,lu1:


if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and (P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6}) ) then
  print(`Bad input`):
  RETURN(FAIL):
fi:

gu:=P[1][1]^(P[1][2]*n1):

gu:=gu*Lc(P[2][1]*n1)^P[2][2]:
lu:=P[3]:

for i from 1 to nops(lu) do
lu1:=lu[i]:
gu:=gu*PpG(lu1[1],lu1[2],n1,trunc(lu1[3]*n1),trunc(lu1[4]*n1),lu1[5],lu1[6]    ):
od:

gu:

end:




#EvalPpG(PA,n1): evaluates a   PpG [Cn,Dn] where Cn and Dn are partial PpG objects
#For GBC(0,0,0,0) (alias Zeta(2))
#EvalPpG( [ [[1,1],[1,0],[]],    [[1, 1], [1, 2],[] ] ] ,1000  );
EvalPpG:=proc(PA,n1) :
if not (type(PA,list) and nops(PA)=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

if EvalPpG1(PA[1],n1)=FAIL or EvalPpG1(PA[2],n1)=FAIL  then
 RETURN(FAIL):
fi:

 EvalPpG1(PA[2],n1)/EvalPpG1(PA[1],n1):
end:



#AsyPpG1(P): inputs a partial Pp expression  and outputs the
#EXACT value of limit of log( EvalPpG1(P,n))/n as n goes to infinity. Try
#AsyPpG1([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1],  [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ]  );
AsyPpG1:=proc(P) local gu,lu,i:


if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and 
(P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6} )) then
  print(`Bad input`):
  RETURN(FAIL):
fi:


gu:=P[1][2]*log(P[1][1]):


gu:=gu+P[2][1]*P[2][2]:


lu:=P[3]:

for i from 1 to nops(lu) do
gu:=gu+PpGlimit(op(lu[i])):
od:

gu:

end:

#AsyPpG(PA,n1): Inputs a GpP expression in the form [Cn,Dn] where Cn a, Dn are the partial PpG expressions for the sequences Cn and Dn (see paper)
#outputs nu sich that the INTEGERating sequence Dn/Cn is asympotic to exp(nu*n). 
#For GBC(0,0,0,0) (alias Zeta(2))
#AsyPpG(      [[1,1],[1,0],[]],    [[1, 1], [1, 2],[] ] ] ],1000  );
AsyPpG:=proc(PA) :
if not (type(PA,list) and nops(PA)=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

if AsyPpG1(PA[1])=FAIL or AsyPpG1(PA[2])=FAIL  then
 RETURN(FAIL):
fi:

AsyPpG1(PA[2])-AsyPpG1(PA[1]):
end:


#The limit of log PpG(a,b,r1*n,r2*n,C,M)/n as n goes to infinity. Try
#PpGlimit(1/3,2/3,3/2,3,{2},3);
PpGlimit:=proc(a,b,r1,r2,C,M)
(PpLimit(a,b,r2)-PpLimit(a,b,r1))*nops(C)/numtheory[phi](M):
end:


#PrintPpG(P,n,L1,L2): P is the  same input as the P EvalPpG(P,n1) (q.v.) but n,L1 and L2 are symbolcs
#where n stands for n,  L1(n) stands for lcm(1...n) and L2(alpha,beta,a,b,C,M,n) stands for the
#product of all primes between a*n and b*n such that the fractional part of n/p is between alpha and beta
#and p mod M belongs to the set C where C is a subset of the set of nonneg. integers less than M relatively prime to it.
#If no modularity condition is present than M=1 and C={0}. Try:

#PrintPpG([[1, 1], [3, 1],[] ],n,LCM,P );
#PrintPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1],  [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ],n,LCM,P );
#PrintPpG([[2, 6], [4, 1], [ [1/4,3/4,0,2,{0},1],  [3/4,1,0,4/3,{0},1], [3/4,1,0,4/3,{0},1] ] ],n, LCM,P );
PrintPpG:=proc(P,n,L1,L2) local gu,lu,i,lu1:


if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and (P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6}) ) then
  print(`Bad input`):
  RETURN(FAIL):
fi:

gu:=1/P[1][1]^(P[1][2]*n):

gu:=gu*L1(P[2][1]*n)^P[2][2]:
lu:=P[3]:

for i from 1 to nops(lu) do
lu1:=lu[i]:
gu:=gu*L2(op(lu1),n):
od:

gu:

end:


####################
# PpLimit(a, b, r): Return the limit of log(Pp(a, b, n, rn)) / n as n tends to
# infinit.
# Try:
#       evalf([log(Pp(1/4, 3/4, 1000, 2 * 1000)) / 1000, PpLimit(1/4, 3/4, 2)]);
#       evalf([log(Pp(3/4, 1, 1000, 4 * 1000 / 3)) / 1000, PpLimit(3/4, 1, 4/3)]);
#       evalf([log(Pp(1/10, 1/2, 1000, 5 * 1000)) / 1000, PpLimit(1/10, 1/2, 5)]);
#       evalf([log(Pp(1/10, 2/10, 2000, 3 * 2000)) / 2000, PpLimit(1/10, 2/10, 3)]);
PpLimit:=proc(a, b, r)
if r=0 then
 RETURN(0):

elif a=0 and b=1 then
r:
elif a=0 then
r-PpLimit(b, 1, r):

else
    Psi(b) - Psi(a) - ifelse(a < 1 / r, 1 / a - max(r, 1 / b), 0);
fi:
end:

###END PRIMES PROCEDURES
#PrintPpG1(P,n,L1,L2): P is the  same input as the P EvalPpG1(P,n1) (q.v.) but n,L1 and L2 are symbolcs
#where n stands for n,  L1(n) stands for lcm(1...n) and L2(alpha,beta,a,b,C,M,n) stands for the
#product of all primes between a*n and b*n such that the fractional part of n/p is between alpha and beta
#and p mod M belongs to the set C where C is a subset of the set of nonneg. integers less than M relatively prime to it.
#If no modularity condition is present than M=1 and C={0}. Try:
PrintPpG1:=proc(P,n,L1,L2) local gu,lu,i,lu1:


if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and (P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6}) ) then
  print(`Bad input`):
  RETURN(FAIL):
fi:

gu:=P[1][1]^(P[1][2]*n):

gu:=gu*L1(P[2][1]*n)^P[2][2]:
lu:=P[3]:

for i from 1 to nops(lu) do
lu1:=lu[i]:
gu:=gu*L2(op(lu1),n):
od:

gu:

end:



#PrintPpG(PA,n,L1,L2): P is the  same input as the P EvalPpG(P,n1) (q.v.) but n,L1 and L2 are symbolcs
#where n stands for n,  L1(n) stands for lcm(1...n) and L2(alpha,beta,a,b,C,M,n) stands for the
#product of all primes between a*n and b*n such that the fractional part of n/p is between alpha and beta
#and p mod M belongs to the set C where C is a subset of the set of nonneg. integers less than M relatively prime to it.
#If no modularity condition is present than M=1 and C={0}. Try:
PrintPpG:=proc(PA,n,L1,L2) :

if not (type(PA,list) and nops(PA)=2) then
print(`Bad input`):
 RETURN(FAIL):	
fi:

if PrintPpG1(PA[1],n,L1,L2)=FAIL or PrintPpG1(PA[2],n,L1,L2)=FAIL then
 RETURN(FAIL):
fi:

PrintPpG1(PA[2],n,L1,L2)/PrintPpG1(PA[1],n,L1,L2):
end:



#WadimHWv(): A verbose version of WadimHW(): lots of conjectured evaluations. Each bunch are probably equivalent to each other
#via some transformation that Wadim Zudilin can figure out. Print:
#WadimHWv():
WadimHWv:=proc() local GU,a1,a2,b1,b2,x,y,W,i:
GU:=WadimHW():

print(`Some Challenges for Wadim Zudlin`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
#print(`Let W(a1,a2,b1,b2) be equal to the following Beukers-type intgeral`):
#print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
#print(`Or equivalently to`):
#print(Sum(RF(-a1+1,n)*RF(-b1+1,n)/(RF(2-a1-a2,n)*RF(2-b1-b2,n)),n=0..infinity)):
#print(``):
#print(`Where RF(a,n) is the rising-factorial a(a+1)*...(a+n-1) `):
print(`As usual let 3F2(a1,a2,a3;b1,b2;1) be the sum of (a1)_n*(a2)_n*(a3)_n/(n!*(b1)_n*(b2)_n),n=0..infinity`):
print(``):

print(`The following bunch of evaluations feature log(2) `):

print(`They are  probably equivalent to each other (and some of them we can ourselves`):

for i from 1 to nops(GU[1]) do
#print(W(op(GU[1][i][1]))=GU[1][i][2]):
print(IntGB3F2(op(GU[1][i][1]))=GU[1][i][2]):
od:

print(`The following bunch of evaluations feature Pi*sqrt(3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[2]) do
print(IntGB3F2(op(GU[2][i][1]))=GU[2][i][2]):
od:


print(`The following bunch of evaluations feature sqrt(2) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[3]) do
print(IntGB3F2(op(GU[3][i][1]))=GU[3][i][2]):
od:



print(`The following bunch of evaluations feature sqrt(5) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[4]) do
print(IntGB3F2(op(GU[4][i][1]))=GU[4][i][2]):
od:




print(`The following bunch of evaluations feature sqrt(3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[5]) do
print(IntGB3F2(op(GU[5][i][1]))=GU[5][i][2]):
od:



print(`The following bunch of evaluations feature the square of the cubic root of 2, i.e. 2^(2/3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[6]) do
print(IntGB3F2(op(GU[6][i][1]))=GU[6][i][2]):
od:



end:





#WadimHWvLP(): A verbose version of WadimHW(): lots of conjectured evaluations. Each bunch are probably equivalent to each other
#via some transformation that Wadim Zudilin can figure out. Print:
#WadimHWvLP():
WadimHWvLP:=proc() local GU,a1,a2,b1,b2,x,y,W,i:
GU:=WadimHW():

print(`Some Challenges for Wadim Zudlin`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
#print(`Let W(a1,a2,b1,b2) be equal to the following Beukers-type intgeral`):
#print( Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(1-a1,1-a2)*Beta(1-b1,1-b2))):
#print(`Or equivalently to`):
#print(Sum(RF(-a1+1,n)*RF(-b1+1,n)/(RF(2-a1-a2,n)*RF(2-b1-b2,n)),n=0..infinity)):
#print(``):
#print(`Where RF(a,n) is the rising-factorial a(a+1)*...(a+n-1) `):
print(`As usual let 3F2(a1,a2,a3;b1,b2;1) be the sum of (a1)_n*(a2)_n*(a3)_n/(n!*(b1)_n*(b2)_n),n=0..infinity`):
print(``):

print(`The following bunch of evaluations feature log(2) `):

print(`They are  probably equivalent to each other (and some of them we can ourselves`):

for i from 1 to nops(GU[1]) do

lprint(IntGB3F2(op(GU[1][i][1]))=GU[1][i][2]):
od:

print(`The following bunch of evaluations feature Pi*sqrt(3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[2]) do
lprint(IntGB3F2(op(GU[2][i][1]))=GU[2][i][2]):
od:


print(`The following bunch of evaluations feature sqrt(2) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[3]) do
lprint(IntGB3F2(op(GU[3][i][1]))=GU[3][i][2]):
od:



print(`The following bunch of evaluations feature sqrt(5) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[4]) do
lprint(IntGB3F2(op(GU[4][i][1]))=GU[4][i][2]):
od:




print(`The following bunch of evaluations feature sqrt(3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[5]) do
lprint(IntGB3F2(op(GU[5][i][1]))=GU[5][i][2]):
od:



print(`The following bunch of evaluations feature the square of the cubic root of 2, i.e. 2^(2/3) `):

print(`They are  probably equivalent to each other `):

for i from 1 to nops(GU[6]) do
lprint(IntGB3F2(op(GU[6][i][1]))=GU[6][i][2]):
od:



end:







#FindRelSym(a1,a2,b1,b2): A clever way to find a relationship
#between IntGB(a1,a2,b1,b2,0) and  IntGB(a1,a2,b1,b2,1)  FOR SYMBOLIC a1,a2,b1,b2. Try:
#FindRelSym(a1,a2,b1,b2);
FindRelSym:=proc(a1,a2,b1,b2) local x,y,c00,c01,c10, d00,d01,d10,gu,eq,var,lu,A,B,C,gu1,gu2,hal,ka,i,j:

option remember:

lu:=x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2):

gu1:=normal(diff((c00+c01*x+c10*y)*x*(1-x)*lu/(1-x*y),x)+diff((d00+d01*x+d10*y)*y*(1-y)*lu/(1-x*y),y)):

gu1:=normal(gu1/lu):

gu2:=normal(A/(1-x*y)+B*x*(1-x)*y*(1-y)/(1-x*y)^2-C):

gu:=normal(gu1-gu2):



gu:=numer(gu):

var:={c00,c01,c10, d00,d01,d10,A,B,C}:

eq:={seq(seq(coeff(coeff(gu,x,i),y,j),j=0..degree(coeff(gu,x,i),y)),i=0..degree(gu,x))}:


var:=solve(eq,var):



hal:=subs(var,[A,B,C]):



if hal[1]=0 then
 RETURN(FAIL):
fi:

hal:=[1,normal(hal[2]/hal[1]),normal(hal[3]/hal[1])]:


[[hal[1],hal[2]],hal[3]]:

end:


#AnBnSym(a1,a2,b1,b2,K): The first K terms in the sequences An and Bn in the expression
#IntGB(a1,a2,b1,b2,n)=B(n)*c-A(n) for SYMBOLIC a1,a2,b1,b2
#where c=IntGB(a1,a2,b1,b2,0); 
#AnBnSym(a1,a2,b1,b2,5);
AnBnSym:=proc(a1,a2,b1,b2,K) local ope,n,N,RE,c0,c1,L,a0,i,gu:

RE:=FindRelSym(a1,a2,b1,b2):
if RE=FAIL then
 RETURN(FAIL):
fi:



c0:=RE[1][1]:
c1:=RE[1][2]:

if c1=0 then
  RETURN(FAIL):
fi:

L:=RE[2]:



ope:=OPEZ2(a1,a2,b1,b2,n,N):

if not IsGoodOpe(ope,n,N)  then
 RETURN(FAIL):
fi:



gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K):



[[seq(-coeff(gu[i],a0,0),i=1..nops(gu))],[seq(coeff(gu[i],a0,1),i=1..nops(gu))]]:

end:

#Info1(a1,a2,b1,b2,K): inputs a four-tuple (a1,a2,b1,b2) and outputs a list consisting of
#(i) the constant as a 3F2, (ii) the floating point to 100 digits,
#(iii) the recurrence operator annihilating An and Bn (iv) the pair of initial conditions for An and Bn
#i.e. [[A(0),A(1)],[B(0),B(1)]] (v) the estimated value of the limit of the log of the INTEGERATING functor divided by n (vi) the
#estimated implied irrationality measure using K values (vii) an identification if successful. Try:
#Info1(0,0,0,0,2000);
Info1:=proc(a1,a2,b1,b2,K) local mu,lu,n,N:

mu:=AnBn(a1,a2,b1,b2,K):
lu:=CnDn(a1,a2,b1,b2,K):

[IntGB3F2(a1,a2,b1,b2),evalf(GBC(a1,a2,b1,b2),100),OPEZ2(a1,a2,b1,b2,n,N),[[op(1..2,mu[1])],[op(1..2,mu[2])]],lu[2][-1],evalf(1+1/lu[3][-1],10),[a1,a2,b1,b2]]:
end:


#Hopefuls5v(K): Some constants that probably have an irrationality proof, indicating those that are not yet identified.
#together for each the relevant information. K should be at least 1000 (it is the parameter used to collect data). Try:
#Hopefuls5v(2000):
Hopefuls5v:=proc(K) local gu,lu,r,ka,ka1:
gu:=Hopefuls5C(r):

print(`This is a collection of candidates for Apery-style proofs using generalizing the Beukers Zeta(2)  integral`):
print(``):

print(`Each entry is a list of length 7 where `):
print(``):
print(`The first entry is the definition of the constant as a 3F2`):
print(``):
print(`The second entry is its value to 100 decimals`):
print(``):
print(`The third entry is the second-order linear recurrence operator in n and N  annihilating Both sequence of rational numbers A(n) and B(n)`):
print(`such that that A(n)/B(n) converges to the constant`):
print(``):
print(`The fourth entry is the pair [[A(0),A[1]],[B(0),B(1)]], of initial conditions enabling the fast computations of A(n) and B(n) using the recurrence`):
print(``):
print(`The fifth entry is the estimate for the limit of log(E(n))/n as n goes to infinity of the INTEGERating factor, i.e. the sequence`):
print(`of rational numbers such that A(n)*E(n) and B(n)*E(n) are BOTH integers`):
print(``):
print(`The sixth entry is the estimate for the implied irrationality measure.`):
print(``):
print(``):
print(`The seventh entry is the quadruple [a1,a2,b1,b2] such that the constant is Int(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(Beta(-a1+1,-b1+1)*B(-a2+1,-b2+1)).`):
print(``):

print(`If Maple can identify the constant, we will mention it.`):
print(``):



print(`We found lots of cases with denominator 2 but they are all (conjecturally) linear-fractionally equaivalent to log(2). Here is one of them`):

lu:=gu[1][1][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,log(2),evalf(log(2)),10000):

print(Info1(op(lu),K)):
print(``):
print(`The value of the constant is `, ka1):


print(`We found lots of cases with denominator 3 but they are all (conjecturally) linear-fractionally equaivalent to sqrt(3)*Pi. Here is one of them`):

lu:=gu[2][1][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,sqrt(3)*Pi,evalf(sqrt(3)*Pi),10000):

print(Info1(op(lu),K)):
print(``):
print(`The value of the constant is probably`, ka1):



print(`We found lots of cases with denominator 4 but they are all (conjecturally) linear-fractionally equaivalent to sqrt(2). Here is one of them`):

lu:=gu[3][1][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,sqrt(2),evalf(sqrt(2)),10000):

print(Info1(op(lu),K)):
print(``):
print(`The value of the constant is probably`, ka1):



print(`We found lots of cases with denominator 5 but they are all (conjecturally) linear-fractionally equaivalent to sqrt(5). Here is one of them`):

lu:=gu[4][1][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,sqrt(5),evalf(sqrt(5)),10000):

print(Info1(op(lu),K)):
print(``):
print(`The value of the constant is probably`, ka1):




print(`We found two cases of denominator 6 where we could identify the constant`):

print(`One of those, that can be expressed in terms of sqrt(3) is`):

lu:=gu[5][2][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,sqrt(3),evalf(sqrt(5)),10000):

print(Info1(op(lu),K)):
print(``):
print(`The value of the constant is probably`, ka1):

print(`------------------------`):

print(`Another one, that can be expressed in terms of the cubic root of 2, is`):

lu:=gu[5][3][1][1]:

ka:=GBC(op(lu)):

ka1:=MyIDs(ka,2^(1/3),evalf(2^(1/3)),10000):

print(Info1(op(lu),K)):

print(``):
print(`The value of the constant is probably`, ka1):

print(`The two classes that are not identified are coming up later.`):

print(`------------------------`):



print(`We found two cases of denominator 7 where we could identify the constant`):

print(`One of those, is a cubic irrationality is`):

lu:=gu[6][2][1][1]:


ka:=GBC(op(lu)):

ka1:=identify(ka):

print(Info1(op(lu),K)):

print(`The constant seems to be`,ka1):



print(`Another one is also a  cubic irrationality is`):

lu:=gu[6][7][1][1]:


ka:=GBC(op(lu)):

ka1:=identify(ka):

print(Info1(op(lu),K)):

print(`The constant seems to be`,ka1):

print(`-----------------------------------------------`):

print(`For the following case,  Maple was unable to identify the constant, so there is hope that we have an irrationality proof of a constant`):
print(`not yet proved to be irrational.`):


print(`Two cases with denominator 6`):

print(Info1(op(gu[5][1][1][1]),K)):

print(Info1(op(gu[5][4][1][1]),K)):



print(`Six cases with denominator 7`):

print(Info1(op(gu[6][1][1][1]),K)):

print(Info1(op(gu[6][3][1][1]),K)):

print(Info1(op(gu[6][4][1][1]),K)):

print(Info1(op(gu[6][5][1][1]),K)):

print(Info1(op(gu[6][6][1][1]),K)):

print(Info1(op(gu[6][8][1][1]),K)):




print(``):

print(``):
print(`-----------------------`):


end: