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Again, it’s important to write this style of programs

length

count8

(define (count8 ls)
  (cond
    ((null? ls) 0)
    ((eqv? (car ls) 8) (add1 (count8 (cdr ls))))
    (else (count8 (cdr ls)))))

count8*

(define (count8* ls)
  (cond
    ((null? ls) 0)
    ;; this is our test for listitude 
    ;; we have a list, and it's car is a list
    ((pair? (car ls)) (+ (count8* (car ls)) (count8* (cdr ls))))
    ((eqv? (car ls) 8) (add1 (count8* (cdr ls))))
    (else (count8* (cdr ls)))))

```racket (count8* '(4 (8 (5 (((8)) 7))) (3 8))) 3 ```

Arithmetic examples.

plus

times

expt

Generalizing

This is not news.

  • Wilhelm Ackermann
  • Gabriel Sudan
(define (phi p m n)
  (cond
    [(zero? p) (+ m n)]
    [(and (zero? n) (zero? (sub1 p))) 0]
    [(and (zero? n) (zero? (sub1 (sub1 p)))) 1]
    [(zero? n) m]
    [else (phi (sub1 p) m (phi p m (sub1 n)))]))

After Ackermann’s publication of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today “the Ackermann function” may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function, is defined as follows for nonnegative integers m and n:

– wiki

(define (ack-peter m n)
  (cond
    [(zero? m) (add1 n)]
    [(zero? n) (ack-peter (sub1 m) 1)]
    [else (ack-peter (sub1 m) (ack-peter m (sub1 n)))]))

Updated: