Message from Evgeni Lozitsky, Sept. 11, 2023

I have been studying recursions of the following form:
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)
I was able to find two types of recursions with period eight and three types of recursions with period twelve.

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

X
Y
Z
(2*A^2+2*A*B+B^2-2*A*X-2*B*Y+2*B*Z)/(2*(A+B-X))
(2*A^3+6*A^2*B+5*A*B^2+2*B^3-2*A^2*X-4*A*B*X-B^2*X-2*A^2*Y-2*A*B*Y-2*B^2*Y+2*A*X*Y-2*A*B*Z+2*B*X*Z)/(2*(A+B-X)*(A+B-Y))
(2*A^4+6*A^3*B+9*A^2*B^2+6*A*B^3+2*B^4-2*A^3*X-4*A^2*B*X-5*A*B^2*X-2*B^3*X-2*A^3*Y-4*A^2*B*Y-A*B^2*Y+2*A^2*X*Y+2*A*B*X*Y+B^2*X*Y-2*B^2*Y^2-2*A^3*Z-4*A^2*B*Z-6*A*B^2*Z-2*B^3*Z+2*A^2*X*Z+2*A*B*X*Z+2*B^2*X*Z+2*A^2*Y*Z+2*A*B*Y*Z+2*B^2*Y*Z-2*A*X*Y*Z)/(2*(A+B-X)*(A+B-Y)*(A+B-Z))
(2*A^4+4*A^3*B+5*A^2*B^2+3*A*B^3+B^4-2*A^3*X-3*A^2*B*X-4*A*B^2*X-2*B^3*X+2*A*B^2*Y+B^3*Y+2*A^2*X*Y+3*A*B*X*Y+2*B^2*X*Y-2*A^2*Y^2-2*A*B*Y^2-2*B^2*Y^2-4*A^3*Z-4*A^2*B*Z-4*A*B^2*Z-B^3*Z+2*A^2*X*Z-A*B*X*Z+2*A^2*Y*Z+2*A*B*Y*Z+B^2*Y*Z-2*A*X*Y*Z-B*X*Y*Z+2*A*Y^2*Z+2*A^2*Z^2+2*B*X*Z^2-2*A*Y*Z^2)/((A+B-Y)*(2*A+B-2*X+2*Y-2*Z)*(A+B-Z))
(2*A^4+4*A^3*B+5*A^2*B^2+3*A*B^3+B^4-4*A^3*X-8*A^2*B*X-8*A*B^2*X-3*B^3*X+2*A^2*X^2+4*A*B*X^2+2*B^2*X^2+2*A^3*Y+3*A^2*B*Y+3*A*B^2*Y+B^3*Y-2*A^2*X*Y-3*A*B*X*Y-2*B^2*Y^2-4*A^3*Z-4*A^2*B*Z-4*A*B^2*Z-B^3*Z+6*A^2*X*Z+6*A*B*X*Z+3*B^2*X*Z-2*A*X^2*Z-2*B*X^2*Z-2*A^2*Y*Z-A*B*Y*Z+B^2*Y*Z+2*A*X*Y*Z+B*X*Y*Z+2*A^2*Z^2-2*A*X*Z^2)/((A-X)*(2*A+B-2*X+2*Y-2*Z)*(A+B-Z))
(2*A^4+4*A^3*B+5*A^2*B^2+3*A*B^3+B^4-4*A^3*X-8*A^2*B*X-8*A*B^2*X-3*B^3*X+2*A^2*X^2+4*A*B*X^2+2*B^2*X^2+2*A*B^2*Y+B^3*Y+2*A^2*X*Y+2*A*B*X*Y+B^2*X*Y-2*A*X^2*Y-2*B*X^2*Y-2*A^2*Y^2-2*A*B*Y^2-2*B^2*Y^2+2*A*X*Y^2+2*B*X*Y^2-2*A^3*Z-3*A^2*B*Z-4*A*B^2*Z-B^3*Z+2*A^2*X*Z+5*A*B*X*Z+3*B^2*X*Z-2*B*X^2*Z+2*A^2*Y*Z+A*B*Y*Z+B^2*Y*Z-2*A*X*Y*Z-B*X*Y*Z)/((A-X)*(A-Y)*(2*A+B-2*X+2*Y-2*Z))
(2*A^4+2*A^3*B+3*A^2*B^2+2*A*B^3+B^4-2*A^3*X-2*A^2*B*X-4*A*B^2*X-2*B^3*X-2*A^3*Y-2*A^2*B*Y+A*B^2*Y+B^3*Y+2*A^2*X*Y+2*A*B*X*Y+2*B^2*X*Y-2*B^2*Y^2-2*A^3*Z-2*A^2*B*Z-3*A*B^2*Z-B^3*Z+2*A^2*X*Z+2*A*B*X*Z+2*B^2*X*Z+2*A^2*Y*Z+2*A*B*Y*Z+B^2*Y*Z-2*A*X*Y*Z-2*B*X*Y*Z)/(2*(A-X)*(A-Y)*(A-Z))
(2*A^3-A*B^2-B^3+2*A*B*X+2*B^2*X-2*A^2*Y-2*A*B*Y-2*B^2*Y-2*A^2*Z+B^2*Z-2*B*X*Z+2*A*Y*Z+2*B*Y*Z)/(2*(A-Y)*(A-Z))
(2*A^2+2*A*B+B^2-2*B*X+2*B*Y-2*A*Z-2*B*Z)/(2*(A-Z))
X
Y
Z
****************************************************************************************************************************************************
A1=A
A2=B
A3=(1+I*Sqrt[3])*B/2
A4=-A^2-(1+I*Sqrt[3])*A*B+B^2
B1=1
B2=0
B3=0
B4=(-2*A+(1-I*Sqrt[3])*B)/2

X
Y
Z
(I*(-2*I*A^2+(-2*I+2*Sqrt[3])*A*B+2*I*B^2+2*I*A*X+2*I*B*Y+(I-Sqrt[3])*B*Z))/(2*A+(-1+I*Sqrt[3])*B-2*X)
(2*I*(-2*I*A^3+2*Sqrt[3]*A^2*B+(4*I-2*Sqrt[3])*A*B^2-2*I*B^3+2*I*A^2*X+(I-Sqrt[3])*A*B*X-2*I*B^2*X+2*I*A^2*Y+(-I-Sqrt[3])*A*B*Y+(-I+Sqrt[3])*B^2*Y-2*I*A*X*Y+2*I*A*B*Z-2*I*B*X*Z))/((2*A+(-1+I*Sqrt[3])*B-2*X)*(2*A+(-1+I*Sqrt[3])*B-2*Y))
(4*I*(-2*I*A^4+(3*I+3*Sqrt[3])*A^3*B+(9*I-5*Sqrt[3])*A^2*B^2+(-9*I-Sqrt[3])*A*B^3+(I+Sqrt[3])*B^4+2*I*A^3*X+(-2*I-2*Sqrt[3])*A^2*B*X+(-4*I+2*Sqrt[3])*A*B^2*X+2*I*B^3*X+2*I*A^3*Y+(-2*I-2*Sqrt[3])*A^2*B*Y+(-8*I+2*Sqrt[3])*A*B^2*Y+(3*I+Sqrt[3])*B^3*Y-2*I*A^2*X*Y+(I+Sqrt[3])*A*B*X*Y+2*I*B^2*X*Y+2*I*B^2*Y^2+2*I*A^3*Z+(-2*I-2*Sqrt[3])*A^2*B*Z+(-3*I+3*Sqrt[3])*A*B^2*Z+2*I*B^3*Z-2*I*A^2*X*Z+(I+Sqrt[3])*A*B*X*Z+(I-Sqrt[3])*B^2*X*Z-2*I*A^2*Y*Z+(I+Sqrt[3])*A*B*Y*Z+(I-Sqrt[3])*B^2*Y*Z+2*I*A*X*Y*Z))/((2*A+(-1+I*Sqrt[3])*B-2*X)*(2*A+(-1+I*Sqrt[3])*B-2*Y)*(2*A+(-1+I*Sqrt[3])*B-2*Z))
(4*I*(-4*I*A^4+(I+5*Sqrt[3])*A^3*B+(11*I-5*Sqrt[3])*A^2*B^2+(-9*I-Sqrt[3])*A*B^3+(I+Sqrt[3])*B^4+(I+Sqrt[3])*A^3*X+(6*I-2*Sqrt[3])*A^2*B*X+(-7*I-Sqrt[3])*A*B^2*X+(I+Sqrt[3])*B^3*X+6*I*A^3*Y+(-3*I-5*Sqrt[3])*A^2*B*Y+(-7*I+5*Sqrt[3])*A*B^2*Y+4*I*B^3*Y+(-I-Sqrt[3])*A^2*X*Y+(-3*I+Sqrt[3])*A*B*X*Y+2*I*B^2*X*Y-2*I*A^2*Y^2+(I+Sqrt[3])*A*B*Y^2+(I-Sqrt[3])*B^2*Y^2+(5*I-Sqrt[3])*A^3*Z+(-I-3*Sqrt[3])*A^2*B*Z+(-4*I+2*Sqrt[3])*A*B^2*Z+2*I*B^3*Z+(-I-Sqrt[3])*A^2*X*Z+(-7*I+Sqrt[3])*A*B*X*Z+(3*I+Sqrt[3])*B^2*X*Z+(-7*I+Sqrt[3])*A^2*Y*Z+(2*I+2*Sqrt[3])*A*B*Y*Z+(I-Sqrt[3])*B^2*Y*Z+(I+Sqrt[3])*A*X*Y*Z+2*I*B*X*Y*Z+2*I*A*Y^2*Z+(-I+Sqrt[3])*A^2*Z^2+2*I*B*X*Z^2+(I-Sqrt[3])*A*Y*Z^2))/((2*A+(-1+I*Sqrt[3])*B-2*Y)*(2*A+(-1+I*Sqrt[3])*B-2*Z)*(4*A+(-1+I*Sqrt[3])*B+(-1+I*Sqrt[3])*X-2*Y+(-1-I*Sqrt[3])*Z))
(4*I*Sqrt[3]*((-2*I+2*Sqrt[3])*A^4+(8*I+2*Sqrt[3])*A^3*B+(-2*I-8*Sqrt[3])*A^2*B^2+(-6*I+4*Sqrt[3])*A*B^3+2*I*B^4+(4*I-2*Sqrt[3])*A^3*X+(-10*I-4*Sqrt[3])*A^2*B*X+(-I+7*Sqrt[3])*A*B^2*X+(3*I-Sqrt[3])*B^3*X-2*I*A^2*X^2+(2*I+2*Sqrt[3])*A*B*X^2+(I-Sqrt[3])*B^2*X^2+(I-Sqrt[3])*A^3*Y+(-3*I-Sqrt[3])*A^2*B*Y+(3*I+5*Sqrt[3])*A*B^2*Y+(2*I-2*Sqrt[3])*B^3*Y+(-I+Sqrt[3])*A^2*X*Y+(3*I+Sqrt[3])*A*B*X*Y-2*Sqrt[3]*B^2*X*Y+(-I-Sqrt[3])*B^2*Y^2+(I-3*Sqrt[3])*A^3*Z+(-5*I-Sqrt[3])*A^2*B*Z+(I+3*Sqrt[3])*A*B^2*Z+(I-Sqrt[3])*B^3*Z+(-3*I+3*Sqrt[3])*A^2*X*Z+(6*I+2*Sqrt[3])*A*B*X*Z-2*Sqrt[3]*B^2*X*Z+2*I*A*X^2*Z+(-I-Sqrt[3])*B*X^2*Z+(-I+Sqrt[3])*A^2*Y*Z+(I+Sqrt[3])*A*B*Y*Z+(-I-Sqrt[3])*B^2*Y*Z+(I-Sqrt[3])*A*X*Y*Z+(-I-Sqrt[3])*B*X*Y*Z+(I+Sqrt[3])*A^2*Z^2+(-I-Sqrt[3])*A*X*Z^2))/((3*I+Sqrt[3])*(A-X)*(2*A+(-1+I*Sqrt[3])*B-2*Z)*(4*A+(-1+I*Sqrt[3])*B+(-1+I*Sqrt[3])*X-2*Y+(-1-I*Sqrt[3])*Z))
-(2*I*((6*I+2*Sqrt[3])*A^4+(6*I-8*Sqrt[3])*A^3*B+(-24*I+2*Sqrt[3])*A^2*B^2+(12*I+6*Sqrt[3])*A*B^3-2*Sqrt[3]*B^4+(-6*I-4*Sqrt[3])*A^3*X+(-12*I+10*Sqrt[3])*A^2*B*X+(21*I+Sqrt[3])*A*B^2*X+(-3*I-3*Sqrt[3])*B^3*X+2*Sqrt[3]*A^2*X^2+(6*I-2*Sqrt[3])*A*B*X^2+(-3*I-Sqrt[3])*B^2*X^2+(-9*I-3*Sqrt[3])*A^3*Y+(-3*I+9*Sqrt[3])*A^2*B*Y+(18*I-4*Sqrt[3])*A*B^2*Y+(-6*I-2*Sqrt[3])*B^3*Y+(9*I+5*Sqrt[3])*A^2*X*Y+(6*I-10*Sqrt[3])*A*B*X*Y+(-9*I+Sqrt[3])*B^2*X*Y-2*Sqrt[3]*A*X^2*Y+(-3*I+Sqrt[3])*B*X^2*Y+(3*I+Sqrt[3])*A^2*Y^2-2*Sqrt[3]*A*B*Y^2+(-3*I+Sqrt[3])*B^2*Y^2+(-3*I-Sqrt[3])*A*X*Y^2+2*Sqrt[3]*B*X*Y^2+(-3*I+Sqrt[3])*A^3*Z+(-3*I+3*Sqrt[3])*A^2*B*Z+(9*I-Sqrt[3])*A*B^2*Z+(-3*I-Sqrt[3])*B^3*Z+(3*I-Sqrt[3])*A^2*X*Z+(6*I-4*Sqrt[3])*A*B*X*Z-6*I*B^2*X*Z+(-3*I+Sqrt[3])*B*X^2*Z+(3*I-Sqrt[3])*A^2*Y*Z-2*Sqrt[3]*A*B*Y*Z+(-3*I+Sqrt[3])*B^2*Y*Z+(-3*I+Sqrt[3])*A*X*Y*Z+2*Sqrt[3]*B*X*Y*Z)/((3*I+Sqrt[3])*(A-X)*(A-Y)*(-4*I*A+(I+Sqrt[3])*B+(I+Sqrt[3])*X+2*I*Y+(I-Sqrt[3])*Z)))
I*(I+Sqrt[3])*((-3*I+Sqrt[3])*A^4+(3*I+Sqrt[3])*A^3*B+(-6*I-6*Sqrt[3])*A^2*B^2+(-3*I+7*Sqrt[3])*A*B^3+(3*I-Sqrt[3])*B^4+(3*I-Sqrt[3])*A^3*X+(-3*I-Sqrt[3])*A^2*B*X+4*Sqrt[3]*A*B^2*X+(3*I-Sqrt[3])*B^3*X+(3*I-Sqrt[3])*A^3*Y+(-3*I-Sqrt[3])*A^2*B*Y+(9*I+5*Sqrt[3])*A*B^2*Y-4*Sqrt[3]*B^3*Y+(-3*I+Sqrt[3])*A^2*X*Y+(3*I+Sqrt[3])*A*B*X*Y-2*Sqrt[3]*B^2*X*Y+(-3*I-Sqrt[3])*B^2*Y^2+(3*I-Sqrt[3])*A^3*Z+(-3*I-Sqrt[3])*A^2*B*Z+(3*I+3*Sqrt[3])*A*B^2*Z-2*Sqrt[3]*B^3*Z+(-3*I+Sqrt[3])*A^2*X*Z+(3*I+Sqrt[3])*A*B*X*Z-2*Sqrt[3]*B^2*X*Z+(-3*I+Sqrt[3])*A^2*Y*Z+(3*I+Sqrt[3])*A*B*Y*Z+(-3*I-Sqrt[3])*B^2*Y*Z+(3*I-Sqrt[3])*A*X*Y*Z+(-3*I-Sqrt[3])*B*X*Y*Z)/(2*(3*I+Sqrt[3])*(A-X)*(A-Y)*(A-Z))
(3*I+Sqrt[3])*((-3*I+Sqrt[3])*A^3+(3*I+3*Sqrt[3])*A^2*B+(3*I-5*Sqrt[3])*A*B^2+(-3*I+Sqrt[3])*B^3-2*Sqrt[3]*A*B*X+(-3*I+Sqrt[3])*B^2*X+(3*I-Sqrt[3])*A^2*Y+(-3*I-Sqrt[3])*A*B*Y+2*Sqrt[3]*B^2*Y+(3*I-Sqrt[3])*A^2*Z+(-3*I-3*Sqrt[3])*A*B*Z+2*Sqrt[3]*B^2*Z+2*Sqrt[3]*B*X*Z+(-3*I+Sqrt[3])*A*Y*Z+(3*I+Sqrt[3])*B*Y*Z)/(12*(A-Y)*(A-Z))
(2*Sqrt[3]*A^2+(6*I+2*Sqrt[3])*A*B-2*Sqrt[3]*B^2-2*Sqrt[3]*B*X+(-3*I-Sqrt[3])*B*Y-2*Sqrt[3]*A*Z+(-3*I+Sqrt[3])*B*Z)/(2*Sqrt[3]*(A-Z))
X
Y
Z
****************************************************************************************************************************************************
A1=A
A2=B
A3=(1-I*Sqrt[3])*B/2
A4=-A^2-(1-I*Sqrt[3])*A*B+B^2
B1=1
B2=0
B3=0
B4=(-2*A+(1+I*Sqrt[3])*B)/2

X
Y
Z
-((I*(2*I*A^2+(2*I+2*Sqrt[3])*A*B-2*I*B^2-2*I*A*X-2*I*B*Y+(-I-Sqrt[3])*B*Z))/(2*A+(-1-I*Sqrt[3])*B-2*X))
-((2*I*(2*I*A^3+2*Sqrt[3]*A^2*B+(-4*I-2*Sqrt[3])*A*B^2+2*I*B^3-2*I*A^2*X+(-I-Sqrt[3])*A*B*X+2*I*B^2*X-2*I*A^2*Y+(I-Sqrt[3])*A*B*Y+(I+Sqrt[3])*B^2*Y+2*I*A*X*Y-2*I*A*B*Z+2*I*B*X*Z))/((2*A+(-1-I*Sqrt[3])*B-2*X)*(2*A+(-1-I*Sqrt[3])*B-2*Y)))-((4*I*(2*I*A^4+(-3*I+3*Sqrt[3])*A^3*B+(-9*I-5*Sqrt[3])*A^2*B^2+(9*I-Sqrt[3])*A*B^3+(-I+Sqrt[3])*B^4-2*I*A^3*X+(2*I-2*Sqrt[3])*A^2*B*X+(4*I+2*Sqrt[3])*A*B^2*X-2*I*B^3*X-2*I*A^3*Y+(2*I-2*Sqrt[3])*A^2*B*Y+(8*I+2*Sqrt[3])*A*B^2*Y+(-3*I+Sqrt[3])*B^3*Y+2*I*A^2*X*Y+(-I+Sqrt[3])*A*B*X*Y-2*I*B^2*X*Y-2*I*B^2*Y^2-2*I*A^3*Z+(2*I-2*Sqrt[3])*A^2*B*Z+(3*I+3*Sqrt[3])*A*B^2*Z-2*I*B^3*Z+2*I*A^2*X*Z+(-I+Sqrt[3])*A*B*X*Z+(-I-Sqrt[3])*B^2*X*Z+2*I*A^2*Y*Z+(-I+Sqrt[3])*A*B*Y*Z+(-I-Sqrt[3])*B^2*Y*Z-2*I*A*X*Y*Z))/((2*A+(-1-I*Sqrt[3])*B-2*X)*(2*A+(-1-I*Sqrt[3])*B-2*Y)*(2*A+(-1-I*Sqrt[3])*B-2*Z)))
-((4*I*(4*I*A^4+(-I+5*Sqrt[3])*A^3*B+(-11*I-5*Sqrt[3])*A^2*B^2+(9*I-Sqrt[3])*A*B^3+(-I+Sqrt[3])*B^4+(-I+Sqrt[3])*A^3*X+(-6*I-2*Sqrt[3])*A^2*B*X+(7*I-Sqrt[3])*A*B^2*X+(-I+Sqrt[3])*B^3*X-6*I*A^3*Y+(3*I-5*Sqrt[3])*A^2*B*Y+(7*I+5*Sqrt[3])*A*B^2*Y-4*I*B^3*Y+(I-Sqrt[3])*A^2*X*Y+(3*I+Sqrt[3])*A*B*X*Y-2*I*B^2*X*Y+2*I*A^2*Y^2+(-I+Sqrt[3])*A*B*Y^2+(-I-Sqrt[3])*B^2*Y^2+(-5*I-Sqrt[3])*A^3*Z+(I-3*Sqrt[3])*A^2*B*Z+(4*I+2*Sqrt[3])*A*B^2*Z-2*I*B^3*Z+(I-Sqrt[3])*A^2*X*Z+(7*I+Sqrt[3])*A*B*X*Z+(-3*I+Sqrt[3])*B^2*X*Z+(7*I+Sqrt[3])*A^2*Y*Z+(-2*I+2*Sqrt[3])*A*B*Y*Z+(-I-Sqrt[3])*B^2*Y*Z+(-I+Sqrt[3])*A*X*Y*Z-2*I*B*X*Y*Z-2*I*A*Y^2*Z+(I+Sqrt[3])*A^2*Z^2-2*I*B*X*Z^2+(-I-Sqrt[3])*A*Y*Z^2))/((2*A+(-1-I*Sqrt[3])*B-2*Y)*(2*A+(-1-I*Sqrt[3])*B-2*Z)*(4*A+(-1-I*Sqrt[3])*B+(-1-I*Sqrt[3])*X-2*Y+(-1+I*Sqrt[3])*Z)))
-((4*I*Sqrt[3]*((2*I+2*Sqrt[3])*A^4+(-8*I+2*Sqrt[3])*A^3*B+(2*I-8*Sqrt[3])*A^2*B^2+(6*I+4*Sqrt[3])*A*B^3-2*I*B^4+(-4*I-2*Sqrt[3])*A^3*X+(10*I-4*Sqrt[3])*A^2*B*X+(I+7*Sqrt[3])*A*B^2*X+(-3*I-Sqrt[3])*B^3*X+2*I*A^2*X^2+(-2*I+2*Sqrt[3])*A*B*X^2+(-I-Sqrt[3])*B^2*X^2+(-I-Sqrt[3])*A^3*Y+(3*I-Sqrt[3])*A^2*B*Y+(-3*I+5*Sqrt[3])*A*B^2*Y+(-2*I-2*Sqrt[3])*B^3*Y+(I+Sqrt[3])*A^2*X*Y+(-3*I+Sqrt[3])*A*B*X*Y-2*Sqrt[3]*B^2*X*Y+(I-Sqrt[3])*B^2*Y^2+(-I-3*Sqrt[3])*A^3*Z+(5*I-Sqrt[3])*A^2*B*Z+(-I+3*Sqrt[3])*A*B^2*Z+(-I-Sqrt[3])*B^3*Z+(3*I+3*Sqrt[3])*A^2*X*Z+(-6*I+2*Sqrt[3])*A*B*X*Z-2*Sqrt[3]*B^2*X*Z-2*I*A*X^2*Z+(I-Sqrt[3])*B*X^2*Z+(I+Sqrt[3])*A^2*Y*Z+(-I+Sqrt[3])*A*B*Y*Z+(I-Sqrt[3])*B^2*Y*Z+(-I-Sqrt[3])*A*X*Y*Z+(I-Sqrt[3])*B*X*Y*Z+(-I+Sqrt[3])*A^2*Z^2+(I-Sqrt[3])*A*X*Z^2))/((-3*I+Sqrt[3])*(A-X)*(2*A+(-1-I*Sqrt[3])*B-2*Z)*(4*A+(-1-I*Sqrt[3])*B+(-1-I*Sqrt[3])*X-2*Y+(-1+I*Sqrt[3])*Z)))
((-I+Sqrt[3])*((6*I+2*Sqrt[3])*A^4+(-9*I+7*Sqrt[3])*A^3*B+(-9*I-13*Sqrt[3])*A^2*B^2+(15*I+3*Sqrt[3])*A*B^3+(-3*I+Sqrt[3])*B^4+(-9*I-Sqrt[3])*A^3*X+(9*I-11*Sqrt[3])*A^2*B*X+(12*I+10*Sqrt[3])*A*B^2*X-6*I*B^3*X+(3*I-Sqrt[3])*A^2*X^2+4*Sqrt[3]*A*B*X^2+(-3*I-Sqrt[3])*B^2*X^2+(-9*I-3*Sqrt[3])*A^3*Y+(12*I-6*Sqrt[3])*A^2*B*Y+(3*I+11*Sqrt[3])*A*B^2*Y+(-6*I-2*Sqrt[3])*B^3*Y+(12*I+2*Sqrt[3])*A^2*X*Y+(-12*I+8*Sqrt[3])*A*B*X*Y+(-3*I-5*Sqrt[3])*B^2*X*Y+(-3*I+Sqrt[3])*A*X^2*Y-2*Sqrt[3]*B*X^2*Y+(3*I+Sqrt[3])*A^2*Y^2+(-3*I+Sqrt[3])*A*B*Y^2-2*Sqrt[3]*B^2*Y^2+(-3*I-Sqrt[3])*A*X*Y^2+(3*I-Sqrt[3])*B*X*Y^2-2*Sqrt[3]*A^3*Z+(3*I-3*Sqrt[3])*A^2*B*Z+(3*I+5*Sqrt[3])*A*B^2*Z+(-3*I-Sqrt[3])*B^3*Z+2*Sqrt[3]*A^2*X*Z+(-3*I+5*Sqrt[3])*A*B*X*Z+(-3*I-3*Sqrt[3])*B^2*X*Z-2*Sqrt[3]*B*X^2*Z+2*Sqrt[3]*A^2*Y*Z+(-3*I+Sqrt[3])*A*B*Y*Z-2*Sqrt[3]*B^2*Y*Z-2*Sqrt[3]*A*X*Y*Z+(3*I-Sqrt[3])*B*X*Y*Z))/((-3*I+Sqrt[3])*(A-X)*(A-Y)*(4*I*A+(-I+Sqrt[3])*B+(-I+Sqrt[3])*X-2*I*Y+(-I-Sqrt[3])*Z))
(1-I*Sqrt[3])*(2*Sqrt[3]*A^4+(-3*I-Sqrt[3])*A^3*B+12*I*A^2*B^2+(-9*I+5*Sqrt[3])*A*B^3-2*Sqrt[3]*B^4-2*Sqrt[3]*A^3*X+(3*I+Sqrt[3])*A^2*B*X+(-6*I+2*Sqrt[3])*A*B^2*X-2*Sqrt[3]*B^3*X-2*Sqrt[3]*A^3*Y+(3*I+Sqrt[3])*A^2*B*Y+(-12*I-2*Sqrt[3])*A*B^2*Y+(6*I-2*Sqrt[3])*B^3*Y+2*Sqrt[3]*A^2*X*Y+(-3*I-Sqrt[3])*A*B*X*Y+(3*I-Sqrt[3])*B^2*X*Y+(3*I+Sqrt[3])*B^2*Y^2-2*Sqrt[3]*A^3*Z+(3*I+Sqrt[3])*A^2*B*Z-6*I*A*B^2*Z+(3*I-Sqrt[3])*B^3*Z+2*Sqrt[3]*A^2*X*Z+(-3*I-Sqrt[3])*A*B*X*Z+(3*I-Sqrt[3])*B^2*X*Z+2*Sqrt[3]*A^2*Y*Z+(-3*I-Sqrt[3])*A*B*Y*Z+(3*I+Sqrt[3])*B^2*Y*Z-2*Sqrt[3]*A*X*Y*Z+(3*I+Sqrt[3])*B*X*Y*Z)
(2*Sqrt[3]*A^3-6*I*A^2*B+(6*I-4*Sqrt[3])*A*B^2+2*Sqrt[3]*B^3+(3*I-Sqrt[3])*A*B*X+2*Sqrt[3]*B^2*X-2*Sqrt[3]*A^2*Y+(3*I+Sqrt[3])*A*B*Y+(-3*I+Sqrt[3])*B^2*Y-2*Sqrt[3]*A^2*Z+6*I*A*B*Z+(-3*I+Sqrt[3])*B^2*Z+(-3*I+Sqrt[3])*B*X*Z+2*Sqrt[3]*A*Y*Z+(-3*I-Sqrt[3])*B*Y*Z)/(2*Sqrt[3]*(A-Y)*(A-Z))/(2*(-3*I+Sqrt[3])*(A-X)*(A-Y)*(A-Z))
((3*I+Sqrt[3])*((-3*I+Sqrt[3])*A^2+(-6*I-2*Sqrt[3])*A*B+(3*I-Sqrt[3])*B^2+(3*I-Sqrt[3])*B*X+(3*I+Sqrt[3])*B*Y+(3*I-Sqrt[3])*A*Z+2*Sqrt[3]*B*Z))/(12*(A-Z))
X
Y
Z
****************************************************************************************************************************************************
It's amazing to watch these giant "complex pythons" curl up and bite their tails :)!

Now it's definitely time for me to start writing the article!

I keep getting the feeling that we're dealing with some kind of algebraic structures...
Group representation?
Lie algebras?
Root systems?

Am I the first to discover recursions with a period of twelve?
This is even surprising!

Thanks for your attention!