maple
I'm starting to think in maple these days. Cool
#begin Maple Code interface(prettyprint=false): f := (x) -> a*x^2 + b*x + c: df := D(f): x := proc(n) if n=0 then x0 else simplify( x(n-1) - f(x(n-1)) / df(x(n-1)) ) fi end: with(combinat, fibonacci): Fib := (n) -> fibonacci(n+1): P1 := (n,x) -> add( binomial(2^n,k) * (Fib(2^n-k-2)) * x^k,k=0..2^n): Q1 := (n,x) -> add( binomial(2^n,k) * (Fib(2^n-k-1)) * x^k,k=0..2^n): a:=1: b:=1: c:=(-1): for n from 0 to 4 do printf("x(%a) - P1(%a)/Q1(%a) = %q\n",n,n,n, simplify(x(n) - (P1(n,x0)/Q1(n,x0)))): end do: #end Maple Code