[waterhouse]

I did this once for a Project Euler question. I called the function "awful-thing". It worked.

  ;this calculates (a+b√sqr)^n, using exponentiation by squaring
  (def awful-thing (n a b sqr)
    (afnwith (n n i a r b it 1 rt 0)
      (if (is n 0)
          (list it rt)
          (even n)
          (self (/ n 2)
                (+ (square i) (* sqr (square r)))
                (* 2 i r)
                it
                rt)
          (self (- n 1)
                i
                r
                (+ (* i it) (* sqr r rt))
                (+ (* i rt) (* r it))))))

  (def fib (n)
    (with (pow2 (expt 2 n)
           hh   (map - (awful-thing n 1 1 5) (awful-thing n 1 -1 5)))
      ;now hh = '(a b) where (fib n) = (a + b*root5)/(root5*pow2)
      ;a should = 0
      (/ (cadr hh)
         pow2)))