Message from Evgeni Lozitsky, Sept. 19, 2023

Part 1.

Let Z[n]=F(Z[n-1],...,Z[n-k]) difference equation.
And let T(X)=(G1*X+G2)/(G3*X+G4) and S(X)=(-G2+G4*X)/(G1-G3*X) inverses of each other elements of the group SL2.
That is T(S(X))=S(T(X))=X.
SL2=SL(2, any field).
We can construct the new difference equation: Z[n]=F'(Z[n-1],...,Z[n-k])=S(F(T(Z[n-1]),...,T(Z[n-k]))).
It is easy to see that if the original difference equation has a periodic solution, the new one also has a periodic solution.
With the same period.

What is a rational difference equation?
It is (loosely speaking) a pair of polynomials.
What is a pair of polynomials?
It is an infinite-dimensional vector space.
Thus we have an infinite-dimensional representation of the group SL2.
That would be great:
Decompose this representation into a direct sum of irreducible representations, if possible.
It would be great to describe orbits.
Understand what common properties the elements of one orbit have in common.
Find the simplest elements of these orbits.

It's a huge program of research...
And I don't think I can handle this program on my own...
Although...
The theories of representation of the groups SL(2,R) and SL(2,C) are absolutely classical theories, and these theories are developed in detail back in the middle of the last century!

Part 2.
Let Z[n]=(A1*Z[n-2]+A2*Z[n-1]+A3)/(B1*Z[n-2]+B2*Z[n-1]+B3)-fractionally linear recursion of order two.
And let Z[n-2]=(-A3-A2*Z[n-1]+B3*Z[n]+B2*Z[n-1]*Z[n])/(A1-B1*Z[n])-reverse recursion.
I will assume that the inverse recursion is also fractionally linear.
This means that B2=0.

Question:
Is it true that all periodic recursions are?

And so we'll assume that:
Z[n]=(A1*Z[n-2]+A2*Z[n-1]+A3)/(B1*Z[n-2]+B3).
F(X,Y)=(A1*X+A2*Y+A3)/(B1*X+B3).
Further, we assume that B1 does not equal 0.
F(X,Y)=(A1*X+A2*Y+A3)/(X+B3).
And finally let's do the transformation:
T(X)=X-B3 and S(X)=X+B3.
Then the recursion takes the form:
F(X,Y)=(A1*X+A2*Y+A3)/X.
Reverse recursion:
G(X,Y)=(-A3-A2*Y)/(A1-X).

Let's consider a recursion whose period is five.

X=U1=V7(-A3-A2*V6)/(A1-V5)
Y=U2=V6=(-A3-A2*V5)/(A1-V4)
(A1*U1+A2*U2+A3)/U1=U3=V5(-A3-A2*V4)/(A1-V3)
(A1*U2+A2*U3+A3)/U2=U4=V4(-A3-A2*V3)/(A1-V2)
(A1*U3+A2*U4+A3)/U3=U5=V3=(-A3-A2*V2)/(A1-V1)
X=U6=V2=X
Y=U7=V1=Y

Here is the code in Wolfram Mathematica:

**************************************************************************
U1=X
U2=Y
U3=Factor[(A1*U1+A2*U2+A3)/U1]
U4=Factor[(A1*U2+A2*U3+A3)/U2]
V1=Y
V2=X
V3=Factor[(-A3-A2*V2)/(A1-V1)]
V4=Factor[(-A3-A2*V3)/(A1-V2)]
CoefficientList[Numerator[Factor[U4-V4]], {X, Y}]
**************************************************************************

All coefficients must be equal to zero.
We get a system of equations:

A1^2*A2*A3=0
A1^2*A2^2-A1*A2*A3=0
-A1*A2^2=0
A1^3*A2+A1^2*A3-A1*A2*A3=0
A1^3-A1^2*A2-A1*A2^2=0
-A1^2+A2^2-A3=0
-A1^2*A2-A1*A3=0
-A1^2+A1*A2-A2^2+A3=0
A1=0

Let's factorize the left sides of the equations (I used the command Factor[]).

A1^2*A2*A3=0
A1*A2*(A1*A2-A3)=0
-A1*A2^2=0
A1*(A1^2*A2+A1*A3-A2*A3)
A1*(A1^2-A1*A2-A2^2)=0
-A1^2+A2^2-A3=0
-A1*(A1*A2+A3)=0
-A1^2+A1*A2-A2^2+A3=0
A1=0

These equations have the unique solution:
{A1=0, A3=A2^2}
As a result we get:
A1=0
A2=A
A3=A^2
B1=1
B2=0
B3=0
F(X,Y)=A*(A+Y)/X

I believe there is no need to explain what this is, it is a generalization of Lynnes difference Equation.
But this is not a final generalization.
Let's do the transformation: T(X)=X-B and S(X)=X+B.
A1=-B
A2=A
A3=A^2+A*B-B^2
B1=1
B2=0
B3=B
F(X,Y)=(A^2+A*B-B^2-B*X+A*Y)/(B+X)

Let's consider a recursion whose period is six.
Here is the code in Wolfram Mathematica:

**************************************************************************
U1=X
U2=Y
U3=(A1*U1+A2*U2+A3)/U1
U4=(A1*U2+A2*U3+A3)/U2
U5=(A1*U3+A2*U4+A3)/U3
V1=Y
V2=X
V3=(-A3-A2*V2)/(A1-V1)
V4=(-A3-A2*V3)/(A1-V2)
CoefficientList[Numerator[Factor[U5-V4]], {X, Y}]
**************************************************************************

All coefficients must be equal to zero.
We get a system of equations:

A1^2*A2^2*A3=0
A1^2*A2^3+A1^3*A3-A1*A2^2*A3+A1*A3^2-A2*A3^2=0
A1^3*A2-A1*A2^3-A1^2*A3+A1*A2*A3-A2^2*A3-A3^2=0
-A1^2*A2-A2*A3=0
A1^3*A2^2+A1^2*A2*A3-A1*A2^2*A3=0
A1^4+A1^3*A2-A1^2*A2^2-A1*A2^3+A1^2*A3-2*A1*A2*A3=0
-A1^3-2*A1^2*A2-A1*A3=0
A1*A2=0
-A1^2*A2^2-A1*A2*A3=0
-A1^3-A1^2*A2-A1*A3+A2*A3=0
A1^2+A1*A2+A3=0

Let's factorize the left sides of the equations:

A1^2*A2^2*A3=0
A1^2*A2^3+A1^3*A3-A1*A2^2*A3+A1*A3^2-A2*A3^2=0
A1^3*A2-A1*A2^3-A1^2*A3+A1*A2*A3-A2^2*A3-A3^2=0
-A2*(A1^2+A3)=0
A1*A2*(A1^2*A2+A1*A3-A2*A3)=0
A1*(A1^3+A1^2*A2-A1*A2^2-A2^3+A1*A3-2*A2*A3)=0
-A1*(A1^2+2*A1*A2+A3)=0
A1*A2=0
-A1*A2*(A1*A2+A3)=0
-A1^3-A1^2*A2-A1*A3+A2*A3=0
A1^2+A1*A2+A3=0

These equations have a two solution:
{A2=0, A3=-A1^2} and {A1=0, A3=0}
As a result we get:

A1=A
A2=0
A3=-A^2
B1=1
B2=0
B3=0
F(X,Y)=A*(-A+X)/X

and

A1=0
A2=A
A3=0
B1=1
B2=0
B3=0
F(X,Y)=A*Y/X

The first recursion is trivial because it is the merger of two recursions of period three.
They are very interesting in themselves, but in a different context.
The second recursion, on the other hand, is very interesting.
Let's do the transformation: T(X)=X-B and S(X)=X+B.

A1=-B
A2=A
A3=A*B-B^2
B1=1
B2=0
B3=B
F(X,Y)=(A*B-B^2-B*X+A*Y)/(B+X)

Let me not examine in detail recursions with period eight and twelve.
The systems of equations are already quite complicated.
But they can be solved!
In solving these systems, I often used the command Solve[,{}].
I will only give the code here.

For recursion with period eight:

**************************************************************************
U1=X
U2=Y
U3=(A1*U1+A2*U2+A3)/U1
U4=(A1*U2+A2*U3+A3)/U2
U5=(A1*U3+A2*U4+A3)/U3
U6=(A1*U4+A2*U5+A3)/U4
V1=Y
V2=X
V3=(-A3-A2*V2)/(A1-V1)
V4=(-A3-A2*V3)/(A1-V2)
V5=(-A3-A2*V4)/(A1-V3)
CoefficientList[Numerator[Factor[U6-V5]], {X, Y}]
**************************************************************************

For recursion with period twelve:

**************************************************************************
U1=X
U2=Y
U3=Factor[(A1*U1+A2*U2+A3)/U1]
U4=Factor[(A1*U2+A2*U3+A3)/U2]
U5=Factor[(A1*U3+A2*U4+A3)/U3]
U6=Factor[(A1*U4+A2*U5+A3)/U4]
U7=Factor[(A1*U5+A2*U6+A3)/U5]
U8=Factor[(A1*U6+A2*U7+A3)/U6]
V1=Y
V2=X
V3=Factor[(-A3-A2*V2)/(A1-V1)]
V4=Factor[(-A3-A2*V3)/(A1-V2)]
V5=Factor[(-A3-A2*V4)/(A1-V3)]
V6=Factor[(-A3-A2*V5)/(A1-V4)]
V7=Factor[(-A3-A2*V6)/(A1-V5)]
CoefficientList[Numerator[Factor[U8-V7]], {X, Y}]
**************************************************************************

You can read about the results in part three of my letter.
I have also considered recursions with periods four, seven, nine, ten, and eleven.
In these cases, there are either no solutions at all or only trivial solutions.

Let Z[n]=(A1*Z[n-3]+A2*Z[n-2]+A3*Z[n-1]+A4)/(B1*Z[n-3]+B2*Z[n-2]+B3*Z[n-1]+B4)-fractionally linear recursion of order three.
And let Z[n-3]=(-A4+B4*Z[n]-A2*Z[n-2]+B2*Z[n-1]*Z[n-2]-A3*Z[n-1]+B3*Z[n]*Z[n-1])/(A1-B1*Z[n])-reverse recursion.
I will assume that the inverse recursion is also fractionally linear.
This means that B2=0 and B3=0.

Question:
Is it true that all periodic recursions are?

And so we'll assume that:
Z[n]=(A1*Z[n-3]+A2*Z[n-2]+A3*Z[n-1]+A4)/(B1*Z[n-3]+B4).
F(X,Y,Z)=(A1*X+A2*Y+A3*Z+A4)/(B1*X+B4).
Further, we assume that B1 does not equal 0.
F(X,Y,Z)=(A1*X+A2*Y+A3*Z+A4)/(X+B4).
And finally let's do the transformation:
T(X)=X-B4 and S(X)=X+B4.
Then the recursion takes the form:
F(X,Y,Z)=(A1*X+A2*Y+A3*Z+A4)/X.
Reverse recursion:
G(X,Y,Z)=(-A4-A2*Z-A3*Y)/(A1-X).

Let's consider a recursion whose period is eight.
Here is the code in Wolfram Mathematica:

**************************************************************************
U1=X
U2=Y
U3=Z
U4=Factor[(A1*U1+A2*U2+A3*U3+A4)/(B1*U1+B2*U2+B3*U3+B4)]
U5=Factor[(A1*U2+A2*U3+A3*U4+A4)/(B1*U2+B2*U3+B3*U4+B4)]
U6=Factor[(A1*U3+A2*U4+A3*U5+A4)/(B1*U3+B2*U4+B3*U5+B4)]
V1=Z
V2=Y
V3=X
V4=Factor[(-V1*(B2*V3+B3*V2+B4)+A2*V3+A3*V2+A4)/(B1*V1-A1)]
V5=Factor[(-V2*(B2*V4+B3*V3+B4)+A2*V4+A3*V3+A4)/(B1*V2-A1)]
V6=Factor[(-V3*(B2*V5+B3*V4+B4)+A2*V5+A3*V4+A4)/(B1*V3-A1)]
CoefficientList[Numerator[Factor[U6-V6]], {X, Y, Z}]
**************************************************************************

We get a rather complicated system of equations:

-A1*A2*A3=0
-A1*A2^2=0
-A1^2+A1*A2+A1*A3-A2*A3+A4=0
-A1^2+A2*A3-A4=0
-A1*(A1*A2+A1*A3+A4)=0
A1*A3*(A1*A2-A3^2)=0
A1*A3*(A1*A2-A3^2)=0
A1*A3*(A1*A2-A1*A3-A4)=0
A1^2*A2^2=0
A1*A2*(A1*A2+A1*A3-A3^2-A4)=0
A1^2*A3^3=0
A1^3-2*A1*A2*A3+A3^3+A1*A4-A2*A4=0
A1^3-A1^2*A2-A1*A2^2-A1^2*A3-A1*A2*A3-A2^2*A3+A1*A3^2+A2*A3^2+A2*A4-A3*A4=0
A1^3-A1^2*A2-A2^3-A1^2*A3+A1*A2*A3+A1*A3^2-A1*A4+A3*A4=0
A1*(A1^2*A2+A1*A2^2+A1^2*A3-A2*A3^2+A1*A4-A2*A4)=0
A1*(A1^2*A2+A1^2*A3-A1*A3^2+A1*A4-A3*A4)=0
-A1*(A1^2*A2*A3-A1*A2*A3^2-A1*A3^3-A1*A2*A4+A3^2*A4)=0
-A1*A3*(A1^2*A2-A1^2*A3-A1*A3^2-A1*A4+A3*A4)=0
-A1^2*(A1*A2*A3^2+A1*A2*A4-A3^2*A4)=0
-A1^2*A2*(A1*A2-A3^2-A4)=0
A1^2*A3*(A1*A3+A4)=0
-A1^3*A3^2*A4=0
-A1^2*A3^2*(A1*A3-A4)=0
-A1^4+A1^3*A2+A1^3*A3+2*A1^2*A2*A3-A1^2*A3^2-A1*A2*A3^2-A1*A3^3-A2^2*A4+A3^2*A4=0
-A1*(A1^3*A2+A1^3*A3-A1^2*A3^2-A1*A2*A3^2+A1^2*A4-A1*A2*A4-A1*A3*A4+A3^2*A4)=0
-A1^2*A3*(A1^2*A3+A1*A4-A3*A4)=0

This system can be solved either in several steps or by using the command Solve[, {A1, A2, A3, A4}].
We obtain two nontrivial solutions:
{A1=0, A2=-A3, A4=-A3^2} and {A1=0, A2=A3, A4=A3^2}.
And these are exactly the two families of recursions with period eight that you already know about.

Let me for recursion with period twelve show only the code:

**************************************************************************
U1=X
U2=Y
U3=Z
U4=Factor[(A1*U1+A2*U2+A3*U3+A4)/(B1*U1+B2*U2+B3*U3+B4)]
U5=Factor[(A1*U2+A2*U3+A3*U4+A4)/(B1*U2+B2*U3+B3*U4+B4)]
U6=Factor[(A1*U3+A2*U4+A3*U5+A4)/(B1*U3+B2*U4+B3*U5+B4)]
V1=Z
V2=Y
V3=X
V4=Factor[(-V1*(B2*V3+B3*V2+B4)+A2*V3+A3*V2+A4)/(B1*V1-A1)]
V5=Factor[(-V2*(B2*V4+B3*V3+B4)+A2*V4+A3*V3+A4)/(B1*V2-A1)]
V6=Factor[(-V3*(B2*V5+B3*V4+B4)+A2*V5+A3*V4+A4)/(B1*V3-A1)]
CoefficientList[Numerator[Factor[U6-V6]], {X, Y, Z}]
**************************************************************************

I have also considered recursions with periods five, six, seven, nine, ten and eleven.
In these cases, there are either no solutions at all or only trivial solutions.

At the end of two months of very intense mental work, two things are obvious to me.
And these two things are mutually exclusive.
Thing one:
I'm good, because I've discovered a lot of wonderful periodic recursions :)!
Thing two:
What I am doing is deeply flawed :)!

Suppose we have a system of polynomial equations.
Suppose further that its size in txt format is several MB.
And suppose finally that this system has simple solutions.
What's that mean?
This means that such a system of equations need not be solved...
It is necessary from the very beginning to act in some other way...

Part 3.
Second order fractional linear recursions.
X, Y, F(X, Y),...
F(X,Y)=(A1*X+A2*Y+A3)/(B1*X+B2*Y+B3)

**************************************************************************
Period 5.

A1=-B
A2=A
A3=A^2+A*B-B^2
B1=1
B2=0
B3=B

X
Y
(A^2+A*B-B^2-B*X+A*Y)/(B+X)
(A^3+2*A^2*B-B^3+A^2*X-B^2*X+A^2*Y-B^2*Y-B*X*Y)/((B+X)*(B+Y))
(A^2+A*B-B^2+A*X-B*Y)/(B+Y)
X
Y
**************************************************************************
Period 6.

A1=-B
A2=A
A3=A*B-B^2
B1=1
B2=0
B3=B

X
Y
(A*B-B^2-B*X+A*Y)/(B+X)
(A^2-B^2-B*X)/(B+X)
(A^2-B^2-B*Y)/(B+Y)
(A*B-B^2+A*X-B*Y)/(B+Y)
X
Y
**************************************************************************
Period 8.

A1=2*(A-B)
A2=-2*I*A
A3=(-1+I)*A^2+2*(1-I)*A*B-2*B^2
B1=2
B2=0
B3=2*B

X
Y
((-1+I)*A^2+(2-2*I)*A*B-2*B^2+2*A*X-2*B*X-2*I*A*Y)/(2*(B+X))
((1+I)*A^3-(3+I)*A^2*B+2*A*B^2-2*B^3-(1+I)*A^2*X+2*A*B*X-2*B^2*X-2*A^2*Y+2*A*B*Y-2*B^2*Y+2*A*X*Y-2*B*X*Y)/(2*(B+X)*(B+Y))
(2*A^3-(6-2*I)*A^2*B+(6-2*I)*A*B^2-4*B^3-2*A^2*X+2*A*B*X-(2+2*I)*B^2*X-(4-2*I)*A^2*Y+(8-4*I)*A*B*Y-(6-2*I)*B^2*Y+2*A*X*Y-(2+2*I)*B*X*Y+(2-2*I)*A*Y^2-(2-2*I)*B*Y^2)/(2*(B+Y)*(-A+2*B+(1+I)*X+(1-I)*Y))
(2*A^3-(6-2*I)*A^2*B+(6-2*I)*A*B^2-4*B^3-2*A^2*X+4*A*B*X-(6+2*I)*B^2*X-(2+2*I)*B*X^2-(2-2*I)*A^2*Y+(2-4*I)*A*B*Y-(2-2*I)*B^2*Y-2*I*A*X*Y-(2-2*I)*B*X*Y)/(2*(-A+B+X)*(-A+2*B+(1+I)*X+(1-I)*Y))
(2*A^3-(5-I)*A^2*B+4*A*B^2-2*B^3-2*A^2*X+2*A*B*X-2*B^2*X-(1-I)*A^2*Y+2*A*B*Y-2*B^2*Y-2*B*X*Y)/(2*(-A+B+X)*(-A+B+Y)))
((-1+I)*A^2+(2-2*I)*A*B-2*B^2-2*I*A*X-2*B*Y)/(2*(-A+B+Y))
X
Y
**************************************************************************
Period 8.

A1=2*(A-B)
A2=2*I*A
A3=(-1-I)*A^2+2*(1+I)*A*B-2*B^2
B1=2
B2=0
B3=2*B

X
Y
((-1-I)*A^2+(2+2*I)*A*B-2*B^2+2*A*X-2*B*X+2*I*A*Y)/(2*(B+X))
((1-I)*A^3-(3-I)*A^2*B+2*A*B^2-2*B^3-(1-I)*A^2*X+2*A*B*X-2*B^2*X-2*A^2*Y+2*A*B*Y-2*B^2*Y+2*A*X*Y-2*B*X*Y)/(2*(B+X)*(B+Y))
(2*A^3-(6+2*I)*A^2*B+(6+2*I)*A*B^2-4*B^3-2*A^2*X+2*A*B*X-(2-2*I)*B^2*X-(4+2*I)*A^2*Y+(8+4*I)*A*B*Y-(6+2*I)*B^2*Y+2*A*X*Y-(2-2*I)*B*X*Y+(2+2*I)*A*Y^2-(2+2*I)*B*Y^2)/(2*(B+Y)*(-A+2*B+(1-I)*X+(1+I)*Y))
(2*A^3-(6+2*I)*A^2*B+(6+2*I)*A*B^2-4*B^3-2*A^2*X+4*A*B*X-(6-2*I)*B^2*X-(2-2*I)*B*X^2-(2+2*I)*A^2*Y+(2+4*I)*A*B*Y-(2+2*I)*B^2*Y+2*I*A*X*Y-(2+2*I)*B*X*Y)/(2*(-A+B+X)*(-A+2*B+(1-I)*X+(1+I)*Y))
(2*A^3-(5+I)*A^2*B+4*A*B^2-2*B^3-2*A^2*X+2*A*B*X-2*B^2*X-(1+I)*A^2*Y+2*A*B*Y-2*B^2*Y-2*B*X*Y)/(2*(-A+B+X)*(-A+B+Y))
((-1-I)*A^2+(2+2*I)*A*B-2*B^2+2*I*A*X-2*B*Y)/(2*(-A+B+Y))
X
Y
**************************************************************************
Period 12.

A1=2*(A-B)
A2=-2*I*A
A3=(-2-Sqrt[3]+I)*A^2+(2-2*I)*A*B-2*B^2
B1=2
B2=0
B3=2*B

X
Y
-((((2-I)+Sqrt[3])*A^2+2*B*(B+X)+2*I*A*((1+I)*B+I*X+Y))/(2*(B+X)))
(((1+2*I)+I*Sqrt[3])*A^3+2*A*(B+X)*(B+Y)-2*B*(B+X)*(B+Y)-A^2*(((4+I)+Sqrt[3])*B+((2+I)+Sqrt[3])*X+2*Y))/(2*(B+X)*(B+Y))
(((1+2*I)+I*Sqrt[3])*A^3-2*B*(B+Y)*((1+I)*B+I*X+Y)+A*(B+Y)*((4+2*I)*B+X-I*Sqrt[3]*X+2*Y)-A^2*((1+I)*((4-I)+Sqrt[3])*B+((2+I)+Sqrt[3])*X+((3+2*I)+I*Sqrt[3])*Y))/((B+Y)*(-I*((2-I)+Sqrt[3])*A+2*((1+I)*B+I*X+Y)))
(2*(((3-2*I)+(2-I)*Sqrt[3])*A^4-2*B*(B+X)*(B+Y)*((1+I)*B+I*X+Y)+A^3*(((-1+10*I)-(2-5*I)*Sqrt[3])*B+2*I*(2+Sqrt[3])*X+2*((1+2*I)+I*Sqrt[3])*Y)-I*A^2*(((8-5*I)+3*Sqrt[3])*B^2-(1+I)*Y*(((2+I)+Sqrt[3])*X+(1+I)*Y)+B*(((4-I)+Sqrt[3])*X+(1-I)*((6-3*I)+Sqrt[3])*Y))+A*(((3+5*I)-(1-I)*Sqrt[3])*B^3+B^2*((-1+6*I)*X-(2-I)*Sqrt[3]*X+(6+2*I)*Y-2*Sqrt[3]*Y)-X*Y*((1+I)*(1+Sqrt[3])*X+(I+Sqrt[3])*Y)-B*(((2-I)+Sqrt[3])*X^2+((-1-I)+(3+I)*Sqrt[3])*X*Y+((-2+I)+Sqrt[3])*Y^2))))/((((2-I)+Sqrt[3])*A+2*I*((1+I)*B+I*X+Y))*(((2-I)+Sqrt[3])*A^2-2*I*(B+X)*(B+Y)+I*A*(((4+I)+Sqrt[3])*B+((2+I)+Sqrt[3])*X+2*Y)))
(2*(((2-I)+Sqrt[3])*A^4-B*(B+X)*((3*I+Sqrt[3])*B+(I+Sqrt[3])*X+2*I*Y)*(B+Y)+A^3*(2*I*((4+I)+Sqrt[3])*B+2*I*((2+I)+Sqrt[3])*X+((2+3*I)+Sqrt[3])*Y)-A^2*(((1+9*I)+(3+I)*Sqrt[3])*B^2+(1+I)*(1+Sqrt[3])*X^2+((3+I)+(3-I)*Sqrt[3])*X*Y+(1-I*Sqrt[3])*Y^2+B*(((3+7*I)+(5+3*I)*Sqrt[3])*X+((3+4*I)+(2-3*I)*Sqrt[3])*Y))+A*(2*(3*I+Sqrt[3])*B^3+(1-I)*(1+Sqrt[3])*X*(X-I*Y)*Y+B^2*(((2+4*I)+4*Sqrt[3])*X-2*I*((-3-I)+Sqrt[3])*Y)+B*(2*(1+Sqrt[3])*X^2+((-1+I)+(1-3*I)*Sqrt[3])*X*Y-(1+I)*(-I+Sqrt[3])*Y^2))))/((2*A+I*(3*I+Sqrt[3])*B-X+I*Sqrt[3]*X-2*Y)*(((2-I)+Sqrt[3])*A^2-2*I*(B+X)*(B+Y)+I*A*(((4+I)+Sqrt[3])*B+((2+I)+Sqrt[3])*X+2*Y)))
-((2*(((3+2*I)+(2+I)*Sqrt[3])*A^4-I*B*(B+X)*(B+Y)*((3+(2+I)*Sqrt[3])*B+((2-I)+Sqrt[3])*X+(1+I)*(1+Sqrt[3])*Y)+A^3*(((-8+6*I)-(6-4*I)*Sqrt[3])*B-2*(3+2*Sqrt[3])*(X+(1-I)*Y))+A*(B+Y)*(2*I*(3+(2+I)*Sqrt[3])*B^2+((2+3*I)+(1+2*I)*Sqrt[3])*X*((1+I)*X+Y)+B*(((-2+13*I)-(3-8*I)*Sqrt[3])*X+(1+I)*((2-I)+Sqrt[3])*Y))-I*A^2*(((11+I)+(7+3*I)*Sqrt[3])*B^2+((2+3*I)+(1+2*I)*Sqrt[3])*X^2+((11+7*I)+(7+5*I)*Sqrt[3])*X*Y+((3-2*I)+(2-I)*Sqrt[3])*Y^2+B*(((12+11*I)+(7+8*I)*Sqrt[3])*X+((15-2*I)+(10+I)*Sqrt[3])*Y))))/((2*A+I*(3*I+Sqrt[3])*B-X+I*Sqrt[3]*X-2*Y)*(2*(2+Sqrt[3])*A^2+I*((2+I)+Sqrt[3])*A*(B+Y)-(1-I)*(1+Sqrt[3])*(B+X)*(B+Y))))
(2*(((-12+7*I)-(7-4*I)*Sqrt[3])*A^4+2*(2+Sqrt[3])*B*(B+X)*(B+Y)*((1+I)*B+I*X+Y)-I*A^3*(((35+8*I)+(20+5*I)*Sqrt[3])*B+((19+5*I)+(11+3*I)*Sqrt[3])*X+2*((7+5*I)+(4+3*I)*Sqrt[3])*Y)+A^2*(((10+25*I)+(5+14*I)*Sqrt[3])*B^2+((2+7*I)+(1+4*I)*Sqrt[3])*X^2+(2+2*I)*((7+5*I)+(4+3*I)*Sqrt[3])*X*Y+((2+7*I)+(1+4*I)*Sqrt[3])*Y^2+B*(((13+33*I)+(7+19*I)*Sqrt[3])*X+((5+30*I)+(2+17*I)*Sqrt[3])*Y))-A*(B+Y)*(((3+13*I)+(1+7*I)*Sqrt[3])*B^2+((2+7*I)+(1+4*I)*Sqrt[3])*X*(X+(1-I)*Y)+B*(2*((4+9*I)+(2+5*I)*Sqrt[3])*X+((6+7*I)+(3+4*I)*Sqrt[3])*Y))))/((2*(2+Sqrt[3])*A^2+I*((2+I)+Sqrt[3])*A*(B+Y)-(1-I)*(1+Sqrt[3])*(B+X)*(B+Y))*(((2+I)+Sqrt[3])*A+(1+I)*(1+Sqrt[3])*((1+I)*B+I*X+Y)))
(((2-3*I)+(1-2*I)*Sqrt[3])*A^3+(1+I)*(1+Sqrt[3])*B*(B+X)*((1+I)*B+I*X+Y)+A^2*(((1+7*I)+(1+5*I)*Sqrt[3])*B+(2+3*I)*X+(1+2*I)*Sqrt[3]*X+(1+5*I)*Y+(1+3*I)*Sqrt[3]*Y)-A*((1+3*I)*(1+Sqrt[3])*B^2+((3+2*I)+(2+I)*Sqrt[3])*X*Y+B*(2*I*(1+Sqrt[3])*X+((4+3*I)+(3+2*I)*Sqrt[3])*Y)))/((A-B-X)*(((2+I)+Sqrt[3])*A+(1+I)*(1+Sqrt[3])*((1+I)*B+I*X+Y)))
((2+2*I)*((2-I)+Sqrt[3])*A^3-4*B*(B+X)*(B+Y)+4*A*B*(2*B+X+Y)-2*A^2*(((6-I)+Sqrt[3])*B+2*X+((2-I)+Sqrt[3])*Y))/(4*(A-B-X)*(A-B-Y))
(((2-I)+Sqrt[3])*A^2+2*I*A*((1+I)*B+X)+2*B*(B+Y))/(2*(A-B-Y))
X
Y
**************************************************************************
Period 12.

A1=2*(A-B)
A2=-2*I*A
A3=(-2+Sqrt[3]+I)*A^2+(2-2*I)*A*B-2*B^2
B1=2
B2=0
B3=2*B

X
Y
(((-2+I)+Sqrt[3])*A^2-2*B*(B+X)+2*A*((1-I)*B+X-I*Y))/(2*(B+X))
(((1+2*I)-I*Sqrt[3])*A^3+A^2*(((-4-I)+Sqrt[3])*B+((-2-I)+Sqrt[3])*X-2*Y)+2*A*(B+X)*(B+Y)-2*B*(B+X)*(B+Y))/(2*(B+X)*(B+Y))
(((1+2*I)-I*Sqrt[3])*A^3-2*B*(B+Y)*((1+I)*B+I*X+Y)+A*(B+Y)*((4+2*I)*B+X+I*Sqrt[3]*X+2*Y)+A^2*((1+I)*((-4+I)+Sqrt[3])*B+((-2-I)+Sqrt[3])*X+I*((-2+3*I)+Sqrt[3])*Y))/((B+Y)*(I*((-2+I)+Sqrt[3])*A+2*((1+I)*B+I*X+Y)))
(2*(((3-2*I)-(2-I)*Sqrt[3])*A^4-2*B*(B+X)*(B+Y)*((1+I)*B+I*X+Y)+A^3*(((-1+10*I)+(2-5*I)*Sqrt[3])*B-2*I*(-2+Sqrt[3])*X+2*((1+2*I)-I*Sqrt[3])*Y)+A^2*(I*((-8+5*I)+3*Sqrt[3])*B^2+(1-I)*((-2-I)+Sqrt[3])*X*Y-2*Y^2+B*(I*((-4+I)+Sqrt[3])*X+(1+I)*((-6+3*I)+Sqrt[3])*Y))+A*(((3+5*I)+(1-I)*Sqrt[3])*B^3+X*Y*((1+I)*(-1+Sqrt[3])*X+(-I+Sqrt[3])*Y)+B^2*(((-1+6*I)+(2-I)*Sqrt[3])*X+2*((3+I)+Sqrt[3])*Y)+B*(((-2+I)+Sqrt[3])*X^2+((2-I)+Sqrt[3])*Y^2+(1+I)*X*(Y+(2-I)*Sqrt[3]*Y)))))/((((-2+I)+Sqrt[3])*A+2*((1-I)*B+X-I*Y))*(((-2+I)+Sqrt[3])*A^2+I*A*(((-4-I)+Sqrt[3])*B+((-2-I)+Sqrt[3])*X-2*Y)+2*I*(B+X)*(B+Y)))
(2*(((-2+I)+Sqrt[3])*A^4-B*(B+X)*((-3*I+Sqrt[3])*B+(-I+Sqrt[3])*X-2*I*Y)*(B+Y)+A^3*(2*I*((-4-I)+Sqrt[3])*B+2*I*((-2-I)+Sqrt[3])*X+((-2-3*I)+Sqrt[3])*Y)+A^2*(((1+9*I)-(3+I)*Sqrt[3])*B^2-(1+I)*(-1+Sqrt[3])*X^2+((3+I)-(3-I)*Sqrt[3])*X*Y+(1+I*Sqrt[3])*Y^2+B*(((3+7*I)-(5+3*I)*Sqrt[3])*X+((3+4*I)-(2-3*I)*Sqrt[3])*Y))+A*(2*(-3*I+Sqrt[3])*B^3+(1-I)*(-1+Sqrt[3])*X*(X-I*Y)*Y+B^2*(((-2-4*I)+4*Sqrt[3])*X-2*I*((3+I)+Sqrt[3])*Y)+(1-I)*B*((1+I)*(-1+Sqrt[3])*X^2+(1-I*Sqrt[3])*Y^2+X*(Y+(2-I)*Sqrt[3]*Y)))))/((2*A+(-3-I*Sqrt[3])*B-X-I*Sqrt[3]*X-2*Y)*(((-2+I)+Sqrt[3])*A^2+I*A*(((-4-I)+Sqrt[3])*B+((-2-I)+Sqrt[3])*X-2*Y)+2*I*(B+X)*(B+Y)))
-((2*(((-3-2*I)+(2+I)*Sqrt[3])*A^4+B*(B+X)*(B+Y)*((3*I+(1-2*I)*Sqrt[3])*B+((1+2*I)-I*Sqrt[3])*X+(1-I)*(-1+Sqrt[3])*Y)+A^3*(((8-6*I)-(6-4*I)*Sqrt[3])*B-2*(-3+2*Sqrt[3])*(X+(1-I)*Y))+A*(B+Y)*(2*I*(-3+(2+I)*Sqrt[3])*B^2+((-2-3*I)+(1+2*I)*Sqrt[3])*X*((1+I)*X+Y)+B*(((2-13*I)-(3-8*I)*Sqrt[3])*X+(1+I)*((-2+I)+Sqrt[3])*Y))+A^2*(((-1+11*I)+(3-7*I)*Sqrt[3])*B^2+((-3+2*I)+(2-I)*Sqrt[3])*X^2+((-7+11*I)+(5-7*I)*Sqrt[3])*X*Y+((2+3*I)-(1+2*I)*Sqrt[3])*Y^2+B*(((-11+12*I)+(8-7*I)*Sqrt[3])*X+((2+15*I)+(1-10*I)*Sqrt[3])*Y))))/((2*A+(-3-I*Sqrt[3])*B-X-I*Sqrt[3]*X-2*Y)*(2*(-2+Sqrt[3])*A^2+I*((-2-I)+Sqrt[3])*A*(B+Y)-(1-I)*(-1+Sqrt[3])*(B+X)*(B+Y))))
(2*(((-12+7*I)+(7-4*I)*Sqrt[3])*A^4-2*(-2+Sqrt[3])*B*(B+X)*(B+Y)*((1+I)*B+I*X+Y)+A^3*(((8-35*I)-(5-20*I)*Sqrt[3])*B+((5-19*I)-(3-11*I)*Sqrt[3])*X+2*((5-7*I)-(3-4*I)*Sqrt[3])*Y)+A^2*(((10+25*I)-(5+14*I)*Sqrt[3])*B^2+((2+7*I)-(1+4*I)*Sqrt[3])*X^2+((4+24*I)-(2+14*I)*Sqrt[3])*X*Y+((2+7*I)-(1+4*I)*Sqrt[3])*Y^2+B*(((13+33*I)-(7+19*I)*Sqrt[3])*X+((5+30*I)-(2+17*I)*Sqrt[3])*Y))+A*(B+Y)*(((-3-13*I)+(1+7*I)*Sqrt[3])*B^2+((-2-7*I)+(1+4*I)*Sqrt[3])*X*(X+(1-I)*Y)+B*(2*((-4-9*I)+(2+5*I)*Sqrt[3])*X+((-6-7*I)+(3+4*I)*Sqrt[3])*Y))))/((2*(-2+Sqrt[3])*A^2+I*((-2-I)+Sqrt[3])*A*(B+Y)-(1-I)*(-1+Sqrt[3])*(B+X)*(B+Y))*(((-2-I)+Sqrt[3])*A+(1+I)*(-1+Sqrt[3])*((1+I)*B+I*X+Y)))
(((-2+3*I)+(1-2*I)*Sqrt[3])*A^3+(1+I)*(-1+Sqrt[3])*B*(B+X)*((1+I)*B+I*X+Y)+A^2*(((-1-7*I)+(1+5*I)*Sqrt[3])*B-(2+3*I)*X+(1+2*I)*Sqrt[3]*X-(1+5*I)*Y+(1+3*I)*Sqrt[3]*Y)+A*((-1-3*I)*(-1+Sqrt[3])*B^2+((3+2*I)-(2+I)*Sqrt[3])*X*Y+B*(-2*I*(-1+Sqrt[3])*X+((4+3*I)-(3+2*I)*Sqrt[3])*Y)))/((A-B-X)*(((-2-I)+Sqrt[3])*A+(1+I)*(-1+Sqrt[3])*((1+I)*B+I*X+Y)))
((-1-I)*((-2+I)+Sqrt[3])*A^3-2*B*(B+X)*(B+Y)+2*A*B*(2*B+X+Y)+A^2*(((-6+I)+Sqrt[3])*B-2*X+((-2+I)+Sqrt[3])*Y))/(2*(A-B-X)*(A-B-Y))
(-2*((-2+I)+Sqrt[3])*A^2+4*I*A*((1+I)*B+X)+4*B*(B+Y))/(4*(A-B-Y))
X
Y
**************************************************************************
Period 12.

A1=2*(A-B)
A2=2*I*A
A3=(-2-Sqrt[3]-I*)*A^2+(2+2*I)*A*B-2*B^2*
B1=2
B2=0
B3=2*B

X
Y
-((((2+I)+Sqrt[3])*A^2+2*B*(B+X)-2*A*((1+I)*B+X+I*Y))/(2*(B+X)))
-((I*((2+I)+Sqrt[3])*A^3-2*A*(B+X)*(B+Y)+2*B*(B+X)*(B+Y)+A^2*(((4-I)+Sqrt[3])*B+((2-I)+Sqrt[3])*X+2*Y))/(2*(B+X)*(B+Y)))
(-((2+I)+Sqrt[3])*A^3+2*B*((1+I)*B+X+I*Y)*(B+Y)-A*((2+4*I)*B-(-I+Sqrt[3])*X+2*I*Y)*(B+Y)+A^2*((1+I)*((4+I)+Sqrt[3])*B+((1+2*I)+I*Sqrt[3])*X+((2+3*I)+Sqrt[3])*Y))/((((2+I)+Sqrt[3])*A-2*((1+I)*B+X+I*Y))*(B+Y))
(2*(((3+2*I)+(2+I)*Sqrt[3])*A^4+2*I*B*(B+X)*((1+I)*B+X+I*Y)*(B+Y)-A^3*(((1+10*I)+(2+5*I)*Sqrt[3])*B+2*I*((2+Sqrt[3])*X+((2+I)+Sqrt[3])*Y))+I*A^2*(((8+5*I)+3*Sqrt[3])*B^2+(1+I)*Y*(((1+2*I)+I*Sqrt[3])*X+(1+I)*Y)+B*(((4+I)+Sqrt[3])*X+(1+I)*((6+3*I)+Sqrt[3])*Y))-A*((1+I)*((1+4*I)+Sqrt[3])*B^3+B^2*(((1+6*I)+(2+I)*Sqrt[3])*X+2*((-3+I)+Sqrt[3])*Y)+X*Y*((1-I)*(1+Sqrt[3])*X+(-I+Sqrt[3])*Y)+B*(((2+I)+Sqrt[3])*X^2+(1-I)*(-1+(2+I)*Sqrt[3])*X*Y+((-2-I)+Sqrt[3])*Y^2))))/((((2+I)+Sqrt[3])*A-2*((1+I)*B+X+I*Y))*(((2+I)+Sqrt[3])*A^2+2*I*(B+X)*(B+Y)-I*A*(((4-I)+Sqrt[3])*B+((2-I)+Sqrt[3])*X+2*Y)))
(2*(((2+I)+Sqrt[3])*A^4-B*(B+X)*((-3*I+Sqrt[3])*B+(-I+Sqrt[3])*X-2*I*Y)*(B+Y)+A^3*(-2*I*((4-I)+Sqrt[3])*B-2*I*((2-I)+Sqrt[3])*X+((2-3*I)+Sqrt[3])*Y)+I*A^2*(((9+I)+(1+3*I)*Sqrt[3])*B^2+(1+I)*(1+Sqrt[3])*X^2+((1+3*I)-(1-3*I)*Sqrt[3])*X*Y-(-I+Sqrt[3])*Y^2+B*(((7+3*I)+(3+5*I)*Sqrt[3])*X+((4+3*I)-(3-2*I)*Sqrt[3])*Y))+A*(2*(-3*I+Sqrt[3])*B^3+(1+I)*(1+Sqrt[3])*X*(X+I*Y)*Y+B^2*(((2-4*I)+4*Sqrt[3])*X+2*I*((-3+I)+Sqrt[3])*Y)+B*(2*(1+Sqrt[3])*X^2+(1+I)*(-1+(2+I)*Sqrt[3])*X*Y-(1-I)*(I+Sqrt[3])*Y^2))))/((2*A+(-3-I*Sqrt[3])*B-X-I*Sqrt[3]*X-2*Y)*(((2+I)+Sqrt[3])*A^2+2*I*(B+X)*(B+Y)-I*A*(((4-I)+Sqrt[3])*B+((2-I)+Sqrt[3])*X+2*Y)))
(2*(((-3+2*I)-(2-I)*Sqrt[3])*A^4-B*(B+X)*(B+Y)*((3*I+(1+2*I)*Sqrt[3])*B+I*((2+I)+Sqrt[3])*X+(1+I)*(1+Sqrt[3])*Y)+A^3*(((8+6*I)+(6+4*I)*Sqrt[3])*B+2*(3+2*Sqrt[3])*(X+(1+I)*Y))+A*(B+Y)*((6*I+(2+4*I)*Sqrt[3])*B^2+((3+2*I)+(2+I)*Sqrt[3])*X*((1+I)*X+I*Y)+B*(((2+13*I)+(3+8*I)*Sqrt[3])*X-(1-I)*((2+I)+Sqrt[3])*Y))-A^2*(((1+11*I)+(3+7*I)*Sqrt[3])*B^2+((3+2*I)+(2+I)*Sqrt[3])*X^2+((7+11*I)+(5+7*I)*Sqrt[3])*X*Y+((-2+3*I)-(1-2*I)*Sqrt[3])*Y^2+B*(((11+12*I)+(8+7*I)*Sqrt[3])*X+((-2+15*I)+(1+10*I)*Sqrt[3])*Y))))/((2*A+(-3-I*Sqrt[3])*B-X-I*Sqrt[3]*X-2*Y)*(2*(2+Sqrt[3])*A^2-I*((2-I)+Sqrt[3])*A*(B+Y)-(1+I)*(1+Sqrt[3])*(B+X)*(B+Y)))
-((2*(((12+7*I)+(7+4*I)*Sqrt[3])*A^4+2*I*(2+Sqrt[3])*B*(B+X)*((1+I)*B+X+I*Y)*(B+Y)-A^3*(((8+35*I)+(5+20*I)*Sqrt[3])*B+((5+19*I)+(3+11*I)*Sqrt[3])*X+2*((5+7*I)+(3+4*I)*Sqrt[3])*Y)-A*(B+Y)*((1+I)*((5+8*I)+(3+4*I)*Sqrt[3])*B^2+I*((7+2*I)+(4+I)*Sqrt[3])*X*(X+(1+I)*Y)+I*B*(2*((9+4*I)+(5+2*I)*Sqrt[3])*X+((7+6*I)+(4+3*I)*Sqrt[3])*Y))+I*A^2*(((25+10*I)+(14+5*I)*Sqrt[3])*B^2+((7+2*I)+(4+I)*Sqrt[3])*X^2+((24+4*I)+(14+2*I)*Sqrt[3])*X*Y+((7+2*I)+(4+I)*Sqrt[3])*Y^2+B*(((33+13*I)+(19+7*I)*Sqrt[3])*X+((30+5*I)+(17+2*I)*Sqrt[3])*Y))))/((((2-I)+Sqrt[3])*A-(1+I)*(1+Sqrt[3])*((1+I)*B+X+I*Y))*(2*(2+Sqrt[3])*A^2-I*((2-I)+Sqrt[3])*A*(B+Y)-(1+I)*(1+Sqrt[3])*(B+X)*(B+Y))))
(((2+3*I)+(1+2*I)*Sqrt[3])*A^3-(1+I)*(1+Sqrt[3])*B*(B+X)*((1+I)*B+X+I*Y)-I*A^2*(((7+I)+(5+I)*Sqrt[3])*B+(3+2*I)*X+(2+I)*Sqrt[3]*X+(5+I)*Y+(3+I)*Sqrt[3]*Y)+I*A*((3+I)*(1+Sqrt[3])*B^2+((2+3*I)+(1+2*I)*Sqrt[3])*X*Y+B*(2*(1+Sqrt[3])*X+((3+4*I)+(2+3*I)*Sqrt[3])*Y)))/((A-B-X)*(((2-I)+Sqrt[3])*A-(1+I)*(1+Sqrt[3])*((1+I)*B+X+I*Y)))
((2-2*I)*((2+I)+Sqrt[3])*A^3-4*B*(B+X)*(B+Y)+4*A*B*(2*B+X+Y)-2*A^2*(((6+I)+Sqrt[3])*B+2*X+((2+I)+Sqrt[3])*Y))/(4*(A-B-X)*(A-B-Y))
(((2+I)+Sqrt[3])*A^2-2*A*((1+I)*B+I*X)+2*B*(B+Y))/(2*(A-B-Y))
X
Y
**************************************************************************
Period 12.

A1=2*(A-B)
A2=2*I*A
A3=(-2+Sqrt[3]-I*)*A^2+(2+2*I)*A*B-2*B^2
B1=2
B2=0
B3=2*B

X
Y
(((-2-I)+Sqrt[3])*A^2-2*B*(B+X)+2*A*((1+I)*B+X+I*Y))/(2*(B+X))
(I*((-2-I)+Sqrt[3])*A^3+A^2*(((-4+I)+Sqrt[3])*B+((-2+I)+Sqrt[3])*X-2*Y)+2*A*(B+X)*(B+Y)-2*B*(B+X)*(B+Y))/(2*(B+X)*(B+Y))
(((2+I)-Sqrt[3])*A^3-2*B*((1+I)*B+X+I*Y)*(B+Y)+A*((2+4*I)*B+(I+Sqrt[3])*X+2*I*Y)*(B+Y)+A^2*((1+I)*((-4-I)+Sqrt[3])*B+I*((-2+I)+Sqrt[3])*X+((-2-3*I)+Sqrt[3])*Y))/((((-2-I)+Sqrt[3])*A+2*((1+I)*B+X+I*Y))*(B+Y))
(2*(((3+2*I)-(2+I)*Sqrt[3])*A^4+2*I*B*(B+X)*((1+I)*B+X+I*Y)*(B+Y)+A^2*(((-5+8*I)-3*I*Sqrt[3])*B^2+(1+I)*(((-2+I)+Sqrt[3])*X-(1-I)*Y)*Y+B*(-I*((-4-I)+Sqrt[3])*X+(1-I)*((-6-3*I)+Sqrt[3])*Y))+A^3*(((-1-10*I)+(2+5*I)*Sqrt[3])*B+2*I*((-2+Sqrt[3])*X+((-2-I)+Sqrt[3])*Y))+A*((1+I)*((-1-4*I)+Sqrt[3])*B^3+X*Y*((1-I)*(-1+Sqrt[3])*X+(I+Sqrt[3])*Y)+B^2*(((-1-6*I)+(2+I)*Sqrt[3])*X+2*((3-I)+Sqrt[3])*Y)+B*(((-2-I)+Sqrt[3])*X^2+((2+I)+Sqrt[3])*Y^2+(1-I)*X*(Y+(2+I)*Sqrt[3]*Y)))))/((((-2-I)+Sqrt[3])*A+2*((1+I)*B+X+I*Y))*(((-2-I)+Sqrt[3])*A^2+A*(((1+4*I)-I*Sqrt[3])*B+(1+2*I)*X-I*Sqrt[3]*X+2*I*Y)-2*I*(B+X)*(B+Y)))
(2*(((-2-I)+Sqrt[3])*A^4-B*(B+X)*((3*I+Sqrt[3])*B+(I+Sqrt[3])*X+2*I*Y)*(B+Y)+A^3*(((2+8*I)-2*I*Sqrt[3])*B+((2+4*I)-2*I*Sqrt[3])*X+((-2+3*I)+Sqrt[3])*Y)+A^2*(((1-9*I)-(3-I)*Sqrt[3])*B^2-(1-I)*(-1+Sqrt[3])*X^2+((3-I)-(3+I)*Sqrt[3])*X*Y+(1-I*Sqrt[3])*Y^2+B*(((3-7*I)-(5-3*I)*Sqrt[3])*X-((-3+4*I)+(2+3*I)*Sqrt[3])*Y))+(1+I)*A*(((3+3*I)+(1-I)*Sqrt[3])*B^3+(-1+Sqrt[3])*X*(X+I*Y)*Y+B^2*(((1+3*I)+(2-2*I)*Sqrt[3])*X+(1+I)*((3-I)+Sqrt[3])*Y)+B*((1-I)*(-1+Sqrt[3])*X^2+(1+I*Sqrt[3])*Y^2+X*(Y+(2+I)*Sqrt[3]*Y)))))/((2*A+I*(3*I+Sqrt[3])*B-X+I*Sqrt[3]*X-2*Y)*(((-2-I)+Sqrt[3])*A^2+A*(((1+4*I)-I*Sqrt[3])*B+(1+2*I)*X-I*Sqrt[3]*X+2*I*Y)-2*I*(B+X)*(B+Y)))
(2*(((3-2*I)-(2-I)*Sqrt[3])*A^4-B*(B+X)*(B+Y)*((-3*I+(1+2*I)*Sqrt[3])*B+I*((-2-I)+Sqrt[3])*X+(1+I)*(-1+Sqrt[3])*Y)+A^3*(((-8-6*I)+(6+4*I)*Sqrt[3])*B+2*(-3+2*Sqrt[3])*(X+(1+I)*Y))+A*(B+Y)*((-6*I+(2+4*I)*Sqrt[3])*B^2+((-3-2*I)+(2+I)*Sqrt[3])*X*((1+I)*X+I*Y)+B*(((-2-13*I)+(3+8*I)*Sqrt[3])*X-(1-I)*((-2-I)+Sqrt[3])*Y))-A^2*(((-1-11*I)+(3+7*I)*Sqrt[3])*B^2+((-3-2*I)+(2+I)*Sqrt[3])*X^2+((-7-11*I)+(5+7*I)*Sqrt[3])*X*Y+((2-3*I)-(1-2*I)*Sqrt[3])*Y^2+B*(((-11-12*I)+(8+7*I)*Sqrt[3])*X+((2-15*I)+(1+10*I)*Sqrt[3])*Y))))/((2*A+I*(3*I+Sqrt[3])*B-X+I*Sqrt[3]*X-2*Y)*(2*(-2+Sqrt[3])*A^2+((1+2*I)-I*Sqrt[3])*A*(B+Y)-(1+I)*(-1+Sqrt[3])*(B+X)*(B+Y)))
(2*(((-12-7*I)+(7+4*I)*Sqrt[3])*A^4+2*I*(-2+Sqrt[3])*B*(B+X)*((1+I)*B+X+I*Y)*(B+Y)+A^3*(((8+35*I)-(5+20*I)*Sqrt[3])*B+((5+19*I)-(3+11*I)*Sqrt[3])*X+2*((5+7*I)-(3+4*I)*Sqrt[3])*Y)-A*(B+Y)*(((3-13*I)-(1-7*I)*Sqrt[3])*B^2+I*((-7-2*I)+(4+I)*Sqrt[3])*X*(X+(1+I)*Y)+B*(((8-18*I)-(4-10*I)*Sqrt[3])*X+((6-7*I)-(3-4*I)*Sqrt[3])*Y))+A^2*(((10-25*I)-(5-14*I)*Sqrt[3])*B^2+((2-7*I)-(1-4*I)*Sqrt[3])*X^2+(2+2*I)*((-5-7*I)+(3+4*I)*Sqrt[3])*X*Y+((2-7*I)-(1-4*I)*Sqrt[3])*Y^2+B*(((13-33*I)-(7-19*I)*Sqrt[3])*X+((5-30*I)-(2-17*I)*Sqrt[3])*Y))))/((((-2+I)+Sqrt[3])*A-(1+I)*(-1+Sqrt[3])*((1+I)*B+X+I*Y))*(2*(-2+Sqrt[3])*A^2+((1+2*I)-I*Sqrt[3])*A*(B+Y)-(1+I)*(-1+Sqrt[3])*(B+X)*(B+Y)))
(((-2-3*I)+(1+2*I)*Sqrt[3])*A^3-(1+I)*(-1+Sqrt[3])*B*(B+X)*((1+I)*B+X+I*Y)+A^2*(((-1+7*I)+(1-5*I)*Sqrt[3])*B-(2-3*I)*X+(1-2*I)*Sqrt[3]*X-(1-5*I)*Y+(1-3*I)*Sqrt[3]*Y)+I*A*((3+I)*(-1+Sqrt[3])*B^2+((-2-3*I)+(1+2*I)*Sqrt[3])*X*Y+B*(2*(-1+Sqrt[3])*X+((-3-4*I)+(2+3*I)*Sqrt[3])*Y)))/((A-B-X)*(((-2+I)+Sqrt[3])*A-(1+I)*(-1+Sqrt[3])*((1+I)*B+X+I*Y)))
((-1+I)*((-2-I)+Sqrt[3])*A^3-2*B*(B+X)*(B+Y)+2*A*B*(2*B+X+Y)+A^2*(((-6-I)+Sqrt[3])*B-2*X+((-2-I)+Sqrt[3])*Y))/(2*(A-B-X)*(A-B-Y))
(((4+2*I)-2*Sqrt[3])*A^2-4*A*((1+I)*B+I*X)+4*B*(B+Y))/(4*(A-B-Y))
X
Y
**************************************************************************

Third order fractional linear recursions.
X, Y, Z, F(X,Y,Z),...
F(X,Y,Z)=(A1*X+A2*Y+A3*Z+A4)/(B1*X+B2*Y+B3*Z+B4)

**************************************************************************
Period 8.

A1=A
A2=B
A3=B
A4=-(A^2+2*A*B-B^2)
B1=1
B2=0
B3=0
B4=-A

X
Y
Z
(A^2+2*A*B-B^2-A*X-B*Y-B*Z)/(A-X)
(A^3+A^2*B-3*A*B^2+B^3-A^2*X-A*B*X+B^2*X-A^2*Y+B^2*Y+*A*X*Y-A*B*Z+B^2*Z+B*X*Z)/((A-X)*(A-Y))
(A^4-4*A^2*B^2+4*A*B^3-B^4-A^3*X+2*A*B^2*X-B^3*X-*A^3*Y+4*A*B^2*Y-2*B^3*Y+A^2*X*Y-B^2*X*Y-B^2*Y^2-A^3*Z+*2*A*B^2*Z-B^3*Z+A^2*X*Z-B^2*X*Z+A^2*Y*Z-B^2*Y*Z-*A*X*Y*Z)/((A-X)*(A-Y)*(A-Z))
(A^3+A^2*B-3*A*B^2+B^3-A*B*X+B^2*X-A^2*Y+B^2*Y-A^2*Z-*A*B*Z+B^2*Z+B*X*Z+A*Y*Z)/((A-Y)*(A-Z))
(A^2+2*A*B-B^2-B*X-B*Y-A*Z)/(A-Z)
X
Y
Z

**************************************************************************
Period 8.

A1=A
A2=B
A3=-B
A4=-A^2-B^2
B1=1
B2=0
B3=0
B4=-A

X
Y
Z
(A^2+B^2-A*X-B*Y+B*Z)/(A-X)
(A^3+A^2*B+A*B^2+B^3-A^2*X-A*B*X-B^2*X-A^2*Y-B^2*Y+*A*X*Y-A*B*Z+B^2*Z+B*X*Z)/((A-X)*(A-Y))
(A^4+2*A^2*B^2+B^4-A^3*X-2*A*B^2*X-B^3*X-A^3*Y+A^2*X*Y+*B^2*X*Y-B^2*Y^2-A^3*Z-2*A*B^2*Z+B^3*Z+A^2*X*Z+B^2*X*Z+*A^2*Y*Z+B^2*Y*Z-A*X*Y*Z)/((A-X)*(A-Y)*(A-Z))
(A^3-A^2*B+A*B^2-B^3+A*B*X+B^2*X-A^2*Y-B^2*Y-A^2*Z+*A*B*Z-B^2*Z-B*X*Z+A*Y*Z)/((A-Y)*(A-Z))
(A^2+B^2-B*X+B*Y-A*Z)/(A-Z)
X
Y
Z


**************************************************************************
Period 12.

A1=2*(A-B)
A2=-2*A
A3=2*A
A4=-A^2+2*A*B-2*B^2
B1=2
B2=0
B3=0
B4=2*B

X
Y
Z
-((A^2+2*B*(B+X)-2*A*(B+X-Y+Z))/(2*(B+X)))
(-A^3-2*B*(B+X)*(B+Y)+2*A*(B+X)*(Y-Z)+A^2*(B+X-2*Y+2*Z))/(2*(B+X)*(B+Y))
-((A^4-2*A*(B+X)*(B+Y)*(B+Z)+2*B*(B+X)*(B+Y)*(B+Z)-A^3*(2*B+X-Y+2*Z)+A^2*(3*B^2+2*Y*(-Y+Z)+X*(Y+2*Z)+B*(3*X-Y+4*Z)))/(2*(B+X)*(B+Y)*(B+Z)))
(-A^4-2*B*(B+Y)*(B+Z)*(B+X-Y+Z)+A^3*(3*B+X-Y+3*Z)+A*(B+Z)*(4*B^2+2*Y*(-Y+Z)+X*(Y+2*Z)+B*(3*X+4*Z))-A^2*(5*B^2-2*Y^2+Y*Z+2*Z^2+X*(Y+3*Z)+B*(4*X-2*Y+8*Z)))/((B+Y)*(B+Z)*(-A+2*(B+X-Y+Z)))
(-A^4-2*B*(B+X)*(B+Z)*(B+X-Y+Z)+A^3*(3*B+X-Y+3*Z)+A*(B+Z)*(4*B^2+4*B*X-3*B*Y-X*Y+4*B*Z+2*X*Z)-A^2*(5*B^2-2*Y^2+2*Z^2+X*(Y+3*Z)+B*(4*X-3*Y+8*Z)))/((A-B-X)*(B+Z)*(A-2*(B+X-Y+Z)))
(A^4+2*B*(B+X)*(B+Y)*(B+X-Y+Z)-A^3*(3*B+X-Y+2*Z)+A^2*(5*B^2+Y*(X-2*Y+2*Z)+B*(4*X-2*Y+4*Z))-A*(4*B^3+X*(-2*X+Y)*Z+B^2*(4*X+3*Z)+B*(2*X*Y-2*Y^2-X*Z+3*Y*Z)))/((A-B-X)*(A-B-Y)*(A-2*(B+X-Y+Z)))
(2*A^4+2*B*(B+X)*(B+Y)*(B+Z)-2*A^3*(3*B+X+Z)-2*A*B*(3*B^2+Y*Z+X*(Y+Z)+2*B*(X+Y+Z))+A^2*(9*B^2+Y*(-2*Y+Z)+2*X*(Y+Z)+B*(6*X+Y+5*Z)))/(2*(A-B-X)*(A-B-Y)*(A-B-Z))
(2*A^3-2*B*(B+Y)*(B+Z)-A^2*(5*B+2*Y+Z)+2*A*(3*B^2+X*Z+B*(X+Y+2*Z)))/(2*(A-B-Y)*(A-B-Z))
(A^2-2*A*(B-X+Y)+2*B*(B+Z))/(2*(A-B-Z))
X
Y
Z

**************************************************************************
Period 12.

A1=2*(A-B)
A2=(1-I*Sqrt[3])*A
A3=(-1-I*Sqrt[3])*A
A4=-A^2+2*A*B-2*B^2+(A-2*B)*Sqrt[3]*A*I
B1=2
B2=0
B3=0
B4=2*B

X
Y
Z
(I*(I+Sqrt[3])*A^2-2*B*(B+X)+A*((2-2*I*Sqrt[3])*B+2*X+Y-I*Sqrt[3]*Y-Z-I*Sqrt[3]*Z))/(2*(B+X))
(2*A^3-2*B*(B+X)*(B+Y)+A*(B+X)*((3-I*Sqrt[3])*B+2*Y+Z-I*Sqrt[3]*Z)+A^2*(I*(5*I+Sqrt[3])*B-2*X-2*Y-Z+I*Sqrt[3]*Z))/(2*(B+X)*(B+Y))
((-1-I*Sqrt[3])*A^4-I*A^2*(2*B+X+Y)*((-3*I+Sqrt[3])*B+(-I+Sqrt[3])*Y-2*I*Z)+2*A*(B+X)*(B+Y)*(B+Z)-2*B*(B+X)*(B+Y)*(B+Z)+A^3*((5+3*I*Sqrt[3])*B+X+I*Sqrt[3]*X+2*(Y+I*Sqrt[3]*Y+Z)))/(2*(B+X)*(B+Y)*(B+Z))
((I+Sqrt[3])*(-2*A^4-B*(B+Y)*(B+Z)*(4*B+X+I*Sqrt[3]*X+2*Y+Z-I*Sqrt[3]*Z)+A^3*((9-I*Sqrt[3])*B+2*X+4*Y+3*Z-I*Sqrt[3]*Z)+A*(B+Z)*((8-2*I*Sqrt[3])*B^2+(X+Y)*(2*Y+Z-I*Sqrt[3]*Z)+B*((3-I*Sqrt[3])*X+9*Y-I*Sqrt[3]*Y+2*Z-2*I*Sqrt[3]*Z))+I*A^2*((13*I+3*Sqrt[3])*B^2+2*I*X*Y+2*I*Y^2+3*I*X*Z+Sqrt[3]*X*Z+5*I*Y*Z+Sqrt[3]*Y*Z+I*Z^2+Sqrt[3]*Z^2+B*((5*I+Sqrt[3])*X+(11*I+Sqrt[3])*Y+2*(5*I+2*Sqrt[3])*Z))))/(2*(B+Y)*(B+Z)*(-(I+Sqrt[3])*A+2*(I+Sqrt[3])*B+2*I*X+I*Y+Sqrt[3]*Y-I*Z+Sqrt[3]*Z))
(-(I+Sqrt[3])*A^4-B*(B+X)*(B+Z)*(2*(I+Sqrt[3])*B+2*I*X+I*Y+Sqrt[3]*Y-I*Z+Sqrt[3]*Z)+A^3*((3*I+5*Sqrt[3])*B+(I+Sqrt[3])*X+2*(I*Y+Sqrt[3]*Y+Sqrt[3]*Z))+A*(B+Z)*((I+5*Sqrt[3])*B^2+(-I+Sqrt[3])*X*(Y+Z)+B*(I*X+3*Sqrt[3]*X+2*Sqrt[3]*Y-2*I*Z+2*Sqrt[3]*Z))-A^2*((2*I+8*Sqrt[3])*B^2+X*((I+Sqrt[3])*Y+2*Sqrt[3]*Z)+B*((I+3*Sqrt[3])*X+3*I*Y+5*Sqrt[3]*Y-I*Z+7*Sqrt[3]*Z)+(Y+Z)*((I+Sqrt[3])*Y+(-I+Sqrt[3])*Z)))/((A-B-X)*(B+Z)*((I+Sqrt[3])*A-2*(I+Sqrt[3])*B-I*(2*X+Y-I*Sqrt[3]*Y-Z-I*Sqrt[3]*Z)))
((I+Sqrt[3])*A^4+B*(B+X)*(B+Y)*(2*(I+Sqrt[3])*B+2*I*X+I*Y+Sqrt[3]*Y-I*Z+Sqrt[3]*Z)-A^3*((3*I+5*Sqrt[3])*B+(I+Sqrt[3])*X+2*I*Y+2*Sqrt[3]*Y-I*Z+Sqrt[3]*Z)+A^2*((2*I+8*Sqrt[3])*B^2+B*((I+3*Sqrt[3])*X+4*I*Y+6*Sqrt[3]*Y-5*I*Z+3*Sqrt[3]*Z)+Y*((I+Sqrt[3])*Y+(-I+Sqrt[3])*Z)+X*((I+Sqrt[3])*Y+(-3*I+Sqrt[3])*Z))-A*((I+5*Sqrt[3])*B^3-2*I*X*(X+Y)*Z+B^2*((I+3*Sqrt[3])*X+(3*I+5*Sqrt[3])*Y+2*(-3*I+Sqrt[3])*Z)+B*(Y*((I+Sqrt[3])*Y+(-3*I+Sqrt[3])*Z)+X*(2*(I+Sqrt[3])*Y+(-7*I+Sqrt[3])*Z))))/((A-B-X)*(A-B-Y)*((I+Sqrt[3])*A-2*(I+Sqrt[3])*B-I*(2*X+Y-I*Sqrt[3]*Y-Z-I*Sqrt[3]*Z)))
(2*A^4+A^3*(I*(9*I+Sqrt[3])*B-2*X-3*Y+I*Sqrt[3]*Y-2*Z)+2*B*(B+X)*(B+Y)*(B+Z)-2*A*B*(3*B^2+Y*Z+X*(Y+Z)+2*B*(X+Y+Z))+A^2*(2*(6-I*Sqrt[3])*B^2+(2*X+Y-I*Sqrt[3]*Y)*(Y+Z)+B*(6*X+7*Y-3*I*Sqrt[3]*Y+5*Z-I*Sqrt[3]*Z)))/(2*(A-B-X)*(A-B-Y)*(A-B-Z))
(2*A^3-2*B*(B+Y)*(B+Z)+I*A^2*((5*I+Sqrt[3])*B+2*I*Y+(I+Sqrt[3])*Z)+A*((3-I*Sqrt[3])*B^2+(-1-I*Sqrt[3])*X*Z+B*((-1-I*Sqrt[3])*X+2*Y+Z-I*Sqrt[3]*Z)))/(2*(A-B-Y)*(A-B-Z))
((1-I*Sqrt[3])*A^2+A*(2*I*(I+Sqrt[3])*B-X+I*Sqrt[3]*X+Y+I*Sqrt[3]*Y)+2*B*(B+Z))/(2*(A-B-Z))
X
Y
Z

**************************************************************************
Period 12.

A1=2*(A-B)
A2=(1+I*Sqrt[3])*A
A3=(-1+I*Sqrt[3])*A
A4=-A^2+2*A*B-2*B^2+(-A+2*B)*Sqrt[3]*A*I
B1=2
B2=0
B3=0
B4=2*B

X
Y
Z
((-1-I*Sqrt[3])*A^2-2*B*(B+X)+A*((2+2*I*Sqrt[3])*B+2*X+Y+I*Sqrt[3]*Y-Z+I*Sqrt[3]*Z))/(2*(B+X))
(2*A^3-2*B*(B+X)*(B+Y)+A^2*((-5-I*Sqrt[3])*B-2*X-2*Y-Z-I*Sqrt[3]*Z)+A*(B+X)*((3+I*Sqrt[3])*B+2*Y+Z+I*Sqrt[3]*Z))/(2*(B+X)*(B+Y))
(I*(I+Sqrt[3])*A^4+I*A^2*(2*B+X+Y)*((3*I+Sqrt[3])*B+(I+Sqrt[3])*Y+2*I*Z)+2*A*(B+X)*(B+Y)*(B+Z)-2*B*(B+X)*(B+Y)*(B+Z)+A^3*((5-3*I*Sqrt[3])*B+X-I*Sqrt[3]*X+2*(Y-I*Sqrt[3]*Y+Z)))/(2*(B+X)*(B+Y)*(B+Z))
((I+Sqrt[3])*(I*(I+Sqrt[3])*A^4+B*(B+Y)*(2*I*(I+Sqrt[3])*B+X+I*Sqrt[3]*X-Y+I*Sqrt[3]*Y-2*Z)*(B+Z)+A^3*((6-4*I*Sqrt[3])*B+X-I*Sqrt[3]*X+2*Y-2*I*Sqrt[3]*Y+3*Z-I*Sqrt[3]*Z)+A*(B+Z)*((7-3*I*Sqrt[3])*B^2+(X+Y)*(Y-I*Sqrt[3]*Y+2*Z)+B*((3-I*Sqrt[3])*X+6*Y-4*I*Sqrt[3]*Y+4*Z))+I*A^2*((11*I+5*Sqrt[3])*B^2+I*Y^2+Sqrt[3]*Y^2+4*I*Y*Z+2*Sqrt[3]*Y*Z+2*I*Z^2+B*(2*(2*I+Sqrt[3])*X+7*I*Y+5*Sqrt[3]*Y+11*I*Z+3*Sqrt[3]*Z)+X*((I+Sqrt[3])*Y+(3*I+Sqrt[3])*Z))))/(2*(B+Y)*(B+Z)*(-(-I+Sqrt[3])*A+2*(-I+Sqrt[3])*B-2*I*X-I*Y+Sqrt[3]*Y+I*Z+Sqrt[3]*Z))
(-(-I+Sqrt[3])*A^4-B*(B+X)*(B+Z)*(2*(-I+Sqrt[3])*B-2*I*X-I*Y+Sqrt[3]*Y+I*Z+Sqrt[3]*Z)+A^3*((-3*I+5*Sqrt[3])*B+(-I+Sqrt[3])*X+2*(-I*Y+Sqrt[3]*Y+Sqrt[3]*Z))+A*(B+Z)*((-I+5*Sqrt[3])*B^2+(I+Sqrt[3])*X*(Y+Z)+B*(-I*X+3*Sqrt[3]*X+2*Sqrt[3]*Y+2*I*Z+2*Sqrt[3]*Z))-A^2*((-2*I+8*Sqrt[3])*B^2+X*((-I+Sqrt[3])*Y+2*Sqrt[3]*Z)+B*((-I+3*Sqrt[3])*X-3*I*Y+5*Sqrt[3]*Y+I*Z+7*Sqrt[3]*Z)+(Y+Z)*((-I+Sqrt[3])*Y+(I+Sqrt[3])*Z)))/((A-B-X)*(B+Z)*((-I+Sqrt[3])*A-2*(-I+Sqrt[3])*B+I*(2*X+Y+I*Sqrt[3]*Y-Z+I*Sqrt[3]*Z)))
((-I+Sqrt[3])*A^4+B*(B+X)*(B+Y)*(2*(-I+Sqrt[3])*B-2*I*X-I*Y+Sqrt[3]*Y+I*Z+Sqrt[3]*Z)-A^3*((-3*I+5*Sqrt[3])*B+(-I+Sqrt[3])*X-2*I*Y+2*Sqrt[3]*Y+I*Z+Sqrt[3]*Z)+A^2*((-2*I+8*Sqrt[3])*B^2+B*((-I+3*Sqrt[3])*X-4*I*Y+6*Sqrt[3]*Y+5*I*Z+3*Sqrt[3]*Z)+Y*((-I+Sqrt[3])*Y+(I+Sqrt[3])*Z)+X*((-I+Sqrt[3])*Y+(3*I+Sqrt[3])*Z))-A*((-I+5*Sqrt[3])*B^3+2*I*X*(X+Y)*Z+B^2*((-I+3*Sqrt[3])*X+(-3*I+5*Sqrt[3])*Y+2*(3*I+Sqrt[3])*Z)+B*(Y*((-I+Sqrt[3])*Y+(3*I+Sqrt[3])*Z)+X*(2*(-I+Sqrt[3])*Y+(7*I+Sqrt[3])*Z))))/((A-B-X)*(A-B-Y)*((-I+Sqrt[3])*A-2*(-I+Sqrt[3])*B+I*(2*X+Y+I*Sqrt[3]*Y-Z+I*Sqrt[3]*Z)))
(2*A^4+A^3*((-9-I*Sqrt[3])*B-2*X-3*Y-I*Sqrt[3]*Y-2*Z)+2*B*(B+X)*(B+Y)*(B+Z)-2*A*B*(3*B^2+Y*Z+X*(Y+Z)+2*B*(X+Y+Z))+A^2*(2*(6+I*Sqrt[3])*B^2+(2*X+Y+I*Sqrt[3]*Y)*(Y+Z)+B*(6*X+7*Y+3*I*Sqrt[3]*Y+5*Z+I*Sqrt[3]*Z)))/(2*(A-B-X)*(A-B-Y)*(A-B-Z))
(2*A^3-2*B*(B+Y)*(B+Z)+A^2*((-5-I*Sqrt[3])*B-2*Y-Z-I*Sqrt[3]*Z)+A*((3+I*Sqrt[3])*B^2+I*(I+Sqrt[3])*X*Z+B*(-X+I*Sqrt[3]*X+2*Y+Z+I*Sqrt[3]*Z)))/(2*(A-B-Y)*(A-B-Z))
((1+I*Sqrt[3])*A^2+A*((-2-2*I*Sqrt[3])*B-X-I*Sqrt[3]*X+Y-I*Sqrt[3]*Y)+2*B*(B+Z))/(2*(A-B-Z))
X
Y
Z
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