Computations for Domino Tilings of 2 x n Grids (or Perfect Matchings of Grid Graphs) on Surfaces

Kasteleyn's formula for rectangular grids in the 2 by n case

FullSimplify[2^( n) Product[((Cos[Pi k/(n+1)])^2+(Cos[Pi 1/3])^2)^(1/4),{k,n}] Product[((Cos[Pi k/(n+1)])^2+(Cos[Pi 2/3])^2)^(1/4),{k,n}]]

Table[FullSimplify[(2^n) ((((∏_(k=1)^n)((((1/4)+((Cos[((k π)/(1+n))])^2)))^(1/4))))^2)],{n,1,20}]

Ciucu’s formulae for m by n cylindrical grids

Ciucu’s formula (7.1)

Surd[Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^m)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^m),{k,1,n}],4] Surd[Product[(1+(-Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^m)(1+(-Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^m),{k,1,n}],4]

Ciucu’s formula for even m (5.1)

2^(n-2Floor[n/2]) Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^m)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^m),{k,1,Floor[n/2]}]

Ciucu’s formula for odd m by 2n (7.2)

Product[Product[4(Cos[j Pi/(4r+2)])^2+4(Cos[k Pi/(2n+1)])^2,j,1,2r+1,2],{k,1,n}]

We’ll set m=2 for the first two of these, in order, and do the two calculations of attempting to simplify the formula and then generating a table of values.

FullSimplify[Surd[Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,n}],4] Surd[Product[(1+(-Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(-Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,n}],4]]

FullSimplify[Table[Surd[Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,n}],4] Surd[Product[(1+(-Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(-Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,n}],4],{n,1,5}]]

FullSimplify[2^(n-2Floor[n/2]) Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,Floor[n/2]}]]

FullSimplify[Table[2^(n-2Floor[n/2]) Product[(1+(Cos[k Pi/(1+n)]+Sqrt[1+(Cos[k Pi/(1+n)])^2])^2)(1+(Cos[k Pi/(1+n)]-Sqrt[1+(Cos[k Pi/(1+n)])^2])^2),{k,1,Floor[n/2]}],{n,1,5}]]

And now we will examine each of Ciucu’s formulae for m by 2 grids, in the same order as above.

FullSimplify[Surd[Product[(1+(Cos[k Pi/(1+2)]+Sqrt[1+(Cos[k Pi/(1+2)])^2])^m)(1+(Cos[k Pi/(1+2)]-Sqrt[1+(Cos[k Pi/(1+2)])^2])^m),{k,1,2}],4] Surd[Product[(1+(-Cos[k Pi/(1+2)]+Sqrt[1+(Cos[k Pi/(1+2)])^2])^m)(1+(-Cos[k Pi/(1+2)]-Sqrt[1+(Cos[k Pi/(1+2)])^2])^m),{k,1,2}],4]]

FullSimplify[Sqrt[(1+((((1/2) (-1-(Sqrt[5]))))^m)) (1+((((1/2) (1-(Sqrt[5]))))^m)) (1+((((1/2) (-1+(Sqrt[5]))))^m)) (1+((((1/2) (1+(Sqrt[5]))))^m))]]

FullSimplify[Table[Sqrt[(1+((((1/2) (-1-(Sqrt[5]))))^m)) (1+((((1/2) (1-(Sqrt[5]))))^m)) (1+((((1/2) (-1+(Sqrt[5]))))^m)) (1+((((1/2) (1+(Sqrt[5]))))^m))],{m,1,6}]]

FullSimplify[Table[((((1+((((1/2) (-1-(Sqrt[5]))))^m)) (1+((((1/2) (1-(Sqrt[5]))))^m)) (1+((((1/2) (-1+(Sqrt[5]))))^m)) (1+((((1/2) (1+(Sqrt[5]))))^m)))^(1/4))^2),{m,1,8}]]

FullSimplify[Table[Surd[Product[(1+(Cos[k Pi/(1+2)]+Sqrt[1+(Cos[k Pi/(1+2)])^2])^m)(1+(Cos[k Pi/(1+2)]-Sqrt[1+(Cos[k Pi/(1+2)])^2])^m),{k,1,2}],4] Surd[Product[(1+(-Cos[k Pi/(1+2)]+Sqrt[1+(Cos[k Pi/(1+2)])^2])^m)(1+(-Cos[k Pi/(1+2)]-Sqrt[1+(Cos[k Pi/(1+2)])^2])^m),{k,1,2}],4],{m,1,8}]]

FullSimplify[2^(2-2Floor[2/2]) Product[(1+(Cos[k Pi/(1+2)]+Sqrt[1+(Cos[k Pi/(1+2)])^2])^m)(1+(Cos[k Pi/(1+2)]-Sqrt[1+(Cos[k Pi/(1+2)])^2])^m),{k,1,Floor[2/2]}]]

Expand[(1+((((1/2) (1-(Sqrt[5]))))^m)) (1+((((1/2) (1+(Sqrt[5]))))^m))]

FullSimplify[Table[(1+((((1/2) (1-(Sqrt[5]))))^m)) (1+((((1/2) (1+(Sqrt[5]))))^m)),{m,1,7}]]

For the odd m case, 2n=2 means n=1 for this one. This implies k=1 also.

FullSimplify[Product[4(Cos[j Pi/(4r+2)])^2+4(Cos[Pi/(2 1+1)])^2,{j,1,2r+1,2}]]

FullSimplify[Table[Product[4(Cos[j Pi/(4r+2)])^2+4(Cos[Pi/(2 1+1)])^2,{j,1,2r+1,2}],{r,0,4}]]

Tesler' s Formulae for Mobius odd, even n (m=2), reversing notation from the way he has it listed (i.e where he writes m, we write n and vice-versa) because his twist is on the m side (his notation).

qFib[n_,q_]:= Sum[ Binomial[n-j,j]q^(n-2j),{j,0,Floor[n/2]}]

qFib[n,1]

Re[(1-I)Product[qFib[n,2Cos[(r Pi)/(m+1)]]+qFib[n-2,2Cos[(r Pi)/(m+1)]]+2I(-1)^(r+m/2),{r,1,m/2}]]

FullSimplify[Re[(1-I)Product[qFib[n,2Cos[(r Pi)/(m+1)]]+qFib[n-2,2Cos[(r Pi)/(m+1)]]+2I(-1)^(r+m/2),{r,1,m/2}]]]

Product[qFib[n,2Cos[r Pi/(m+1)]]+qFib[n-2,2Cos[r Pi/(m+1)]],{r,1,m/2}]

FullSimplify[Product[qFib[n,2Cos[r Pi/(m+1)]]+qFib[n-2,2Cos[r Pi/(m+1)]],{r,1,m/2}]]

FullSimplify[Re[(1-I)Product[qFib[n,2Cos[(r Pi)/(2+1)]]+qFib[n-2,2Cos[(r Pi)/(2+1)]]+2I(-1)^(r+2/2),{r,1,2/2}]]]

We want n odd, so substitute 2r+1

FullSimplify[Re[(1-I) (2 I+Fibonacci[1+(2r+1)]+(1/2) (-Fibonacci[(2r+1)]+LucasL[(2r+1)]))]]

Table[Re[(1-I) (2 I+Fibonacci[2+2 r]+(1/2) (-Fibonacci[1+2 r]+LucasL[1+2 r]))],{r,0,15}]

FullSimplify[qFib[n,2Cos[ Pi/(2+1)]]+qFib[n-2,2Cos[ Pi/(2+1)]]]

We want n even here.

Table[Fibonacci[1+n]+(1/2) (-Fibonacci[n]+LucasL[n]),{n,2,20,2}]

If we want the twist on the other side, we do Tesler’s formulae but with n=2.  For m, n even he has as above (copied just below), then for m odd, n even he has the second formula below.

Product[qFib[n,2Cos[r Pi/(m+1)]]+qFib[n-2,2Cos[r Pi/(m+1)]],{r,1,m/2}]

2 Product[qFib[n,2Cos[r Pi/(m+1)]]+qFib[n-2,2Cos[r Pi/(m+1)]],{r,1,(m-1)/2}]

Now let’s set n=2.

FullSimplify[Product[qFib[2,2Cos[r Pi/(m+1)]]+qFib[2-2,2Cos[r Pi/(m+1)]],{r,1,m/2}]]

Table[FullSimplify[(∏_(r=1)^(m/2))(2+4 ((Cos[((π r)/(1+m))])^2))],{m,2,20,2}]

This is A079935 		a(n) = 4*a(n-1) - a(n-2).

FullSimplify[2 Product[qFib[2,2Cos[r Pi/(m+1)]]+qFib[2-2,2Cos[r Pi/(m+1)]],{r,1,(m-1)/2}]]

Table[FullSimplify[2 (∏_(r=1)^((1/2) (-1+m)))(2+4 ((Cos[((π r)/(1+m))])^2))],{m,1,19,2}]

This is A052530 		a(0)=0, a(1)=2; for n>=2, a(n)=4*a(n-1)-a(n-2).

Lu/Wu' s formulae for Mobius odd, even n (m=2)

FullSimplify[Re[(1-I)Product[2I(-1)^3 Sin[(4r-1)Pi/(2n)]+2Cos[Pi/3],{r,1,n}]]]

Table[FullSimplify[Re[(1-I) (∏_(r=1)^n)(1-2 I Sin[((π (-1+4 r))/(2 n))])]],{n,1,10}]

FullSimplify[Product[4(Sin[(4r-1)Pi/(2n)])^2+4(Cos[Pi/3])^2,{r, 1, n/2}]]

Table[FullSimplify[(∏_(r=1)^(n/2))(1+4 ((Sin[((π (-1+4 r))/(2 n))])^2))],{n,2,20,2}]

Lu/Wu' s formulae for Klein odd, even n (m=2)

FullSimplify[Re[(1-I)Product[2I(-1)^3 Sin[(4r-1)Pi/(2n)]+2Sin[Pi/2],{r,1,n}]]]

Table[FullSimplify[Re[(1-I) (∏_(r=1)^n)(2-2 I Sin[((π (-1+4 r))/(2 n))])]],{n,1,10}]

FullSimplify[Product[4(Sin[(4r-1)Pi/(2n)])^2+4(Sin[Pi/2])^2,{r, 1, n/2}]]

Table[FullSimplify[(∏_(r=1)^(n/2))(4+4 ((Sin[((π (-1+4 r))/(2 n))])^2))],{n,2,20,2}]

So there's basically no simplification offered by Mathematica over what we'd do by hand in these cases.

Lu/Wu' s formulae for other-twist Mobius (m=2)

FullSimplify[Product[4(Sin[(3Pi/4)])^2+4(Cos[r Pi/(n+1)])^2,{r,1,(n+1)/2}]]

Table[FullSimplify[(∏_(r=1)^((1+n)/2))(2+4 ((Cos[((π r)/(1+n))])^2))],{n,1,15}]

This is A048788 a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2).

Lu/Wu' s formulae for other-twist Klein (m=2)

FullSimplify[Product[4(Sin[(2r-1)Pi/n])^2+4(Sin[3 Pi/4])^2,{r,1,(n+1)/2}]]

Table[FullSimplify[(∏_(r=1)^((1+n)/2))(2+4 ((Sin[((π (-1+2 r))/n)])^2))],{n,1,10}]

Side-n-twist Mobius bands

RSolve[{a[n]==a[n-1]+a[n-2]+Mod[n,2] a[n-1],a[1]==2,a[2]==3},a[n],n]

FullSimplify[((2^(-(3/2)-2 n)) (((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))) ((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)))))^n) (-5 (Sqrt[3 (2-(Sqrt[3]))]) (((-(2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-5 (Sqrt[3 (2+(Sqrt[3]))]) (((-(2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+5 (2^(1+n)) (Sqrt[3]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-7 (2^(1+n)) (Sqrt[2-(Sqrt[3])]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+7 (2^(1+n)) (Sqrt[2+(Sqrt[3])]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+7 (2^((1/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-5 (Sqrt[3 (2-(Sqrt[3]))]) ((((2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-5 (Sqrt[3 (2+(Sqrt[3]))]) ((((2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-5 (2^(1+n)) (Sqrt[3]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-7 (2^(1+n)) (Sqrt[2-(Sqrt[3])]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+7 (2^(1+n)) (Sqrt[2+(Sqrt[3])]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+(((-1))^(1+n)) (2^((7/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+7 (2^((1/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+(2^(3+n)) (Sqrt[3 (2-(Sqrt[3])) (2+(Sqrt[3]))]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+(((-1))^(1+n)) (2^(3+n)) (Sqrt[3 (2-(Sqrt[3])) (2+(Sqrt[3]))]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])-(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+5 (Sqrt[3 (2-(Sqrt[3]))]) (((-(2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+5 (Sqrt[3 (2+(Sqrt[3]))]) (((-(2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-5 (2^(1+n)) (Sqrt[3]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+7 (2^(1+n)) (Sqrt[2-(Sqrt[3])]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-7 (2^(1+n)) (Sqrt[2+(Sqrt[3])]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)-23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+7 (2^((1/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(3/4)) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+(2^(3+n)) (Sqrt[3 (2-(Sqrt[3])) (2+(Sqrt[3]))]) (((-(1/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+5 (Sqrt[3 (2-(Sqrt[3]))]) ((((2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+5 (Sqrt[3 (2+(Sqrt[3]))]) ((((2/((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))))^n)+5 (2^(1+n)) (Sqrt[3]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+7 (2^(1+n)) (Sqrt[2-(Sqrt[3])]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(3/4)) (((2+(Sqrt[3])))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-7 (2^(1+n)) (Sqrt[2+(Sqrt[3])]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+23 (2^((1/2)+n)) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) (((2-(Sqrt[3])))^(1/4)) (((2+(Sqrt[3])))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^((7/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+(((-1))^(1+n)) (2^((7/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))+7 (2^((1/2)+n)) (Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(3/4)) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))-(2^(3+n)) (Sqrt[3 (2-(Sqrt[3])) (2+(Sqrt[3]))]) ((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])+(Sqrt[2]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4))))^(-n))))/((Sqrt[3]) ((((2-(Sqrt[3])) (2+(Sqrt[3]))))^(1/4)) ((Sqrt[2-(Sqrt[3])])-(Sqrt[2+(Sqrt[3])])) (2 (Sqrt[2-(Sqrt[3])])+(Sqrt[3 (2-(Sqrt[3]))])-2 (Sqrt[2+(Sqrt[3])])+(Sqrt[3 (2+(Sqrt[3]))])))]