# Fibonacci numbers (Scheme)

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The Fibonacci numbers are the integer sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, ..., in which each item is formed by adding the previous two. The sequence can be defined recursively by

1 \\ \end{cases} ."/>

Fibonacci number programs that implement this definition directly are often used as introductory examples of recursion. However, many other algorithms for calculating (or making use of) Fibonacci numbers also exist.

Here we present a few computations of the Fibonacci numbers in Scheme.

## Recursion

(define (fib n)
(if (< n 2)
n
(+ (fib (- n 1)) (fib (- n 2)))))


## Tail Recursive

(define (fib n)
(letrec ((fib-aux (lambda (n a b)
(if (= n 0)
a
(fib-aux (- n 1) b (+ a b))))))
(fib-aux n 0 1)))


## "Clever" Algorithm

Described in Edsger W. Dijkstra's note here.

This algorithm is O(log n) (In practice, it's more like O(n log n))

(define (fib n)
(define (fib-aux a b p q count)
(cond ((= count 0) b)
((even? count)
(fib-aux a
b
(+ (* p p) (* q q))
(+ (* q q) (* 2 p q))
(/ count 2)))
(else
(fib-aux (+ (* b q) (* a q) (* a p))
(+ (* b p) (* a q))
p
q
(- count 1)))))
(fib-aux 1 0 0 1 n))