(define (f n)
  (define v (make-vector (+ 2 n)))
  (vector-set! v 0 4)
  (vector-set! v 1 (/ 17 4))
  (let loop ((i 2))
    (if (<= i n)
        (let ((v1 (vector-ref v (- i 1))) (v2 (vector-ref v (- i 2))))
          (vector-set! v i (+ (- 108 (/ 815 v1)) (/ 1500 (* v1 v2))))
          (loop (+ 1 i)))))
  (vector-ref v n))
(define (p e)
  (print (list e (exact->inexact e))))
(p (f 0))
(p (f 1))
(p (f 2))
(p (f 3))
(p (f 80))

#|
(4 4.0)
(17/4 4.25)
(76/17 4.47058823529412)
(353/76 4.64473684210526)
(206795153138256918939565417139009598365577843034794672964/41359030627651383817474849310671104336332210648235594113 5.0)
|#