[seqfan] Searching for associative rational functions

[Georgi Guninski]

I am searching for associative rational functions mainly for analogues of A175841

Basically by equating coefficients i end up with a nonlinear system - some associative functions found this way are attached (all are dimension 2, degree 2, the sequences are uninteresting to me).

Some questions:

1. Are there faster ways for searching (the system can be large)?
2. Are there analytical limitations to the types of assoc. functions?
3. Does anyone knows/can find degree >=3 associative function - (the system is too large for me)?

Thanks.
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/*test assoc.:*/
/*p1=[x1,x2];p2=[y1,y2];p3=[z1,z2];a1=r(p1,r(p2,p3));a2=r(r(p1,p2),p3);a1==a2;*/


/*v Name : Hexagonal numbers: n(2n-1).*/
{r(a,b)=[a[2]*b[2] + a[1] + a[2] + b[1] + b[2] + 1, a[2] + b[2] + 1]};

/*v [[function= [f(n): f(n + 1) - 6f(n) - 5= 0,f(0)= 8]*/
\\{r(a,b)=[a[1]*b[1] + a[2]*b[1] + a[1]*b[2] + a[2]*b[2] + a[1] + a[2] + b[1] + b[2], a[1]*b[1] + a[2]*b[1] + a[1]*b[2] + a[2]*b[2] + a[1] + a[2] + b[1] + b[2] + 1];}

\\{r(a,b)=[(-6)*a[1]^2 + a[1]*a[2] + a[2]^2 + (-6)*b[1]^2 + b[1]*b[2] + b[2]^2 + 5, (-12)*a[1]^2 + 2*a[1]*a[2] + 2*a[2]^2 + (-12)*b[1]^2 + 2*b[1]*b[2] + 2*b[2]^2 + 51/5]}

\\{r(a,b)=[2*a[1]*a[2] + a[2]^2 + 2*b[1]*b[2] + b[2]^2 ,-4*a[1]*a[2] - 2*a[2]^2 - 4*b[1]*b[2] - 2*b[2]^2 - 1/2]}
{
gg(n,a=[1,1])=if(n%2,n==1&return(a);r(gg(n-1,a),a),a=gg(n\2,a);r(a,a))
}
vector(20,i,gg(i)[1]);/*Hexagonal numbers: n(2n-1)*/