# The Fibonacci sequence

### From HaskellWiki

Implementing the fibonacci sequence is considered the "Hello, world!" of Haskell programming. This page collects Haskell implementations of the sequence.

## Contents |

## 1 Naive solution

fib 0 = 0 fib 1 = 1 fib n = fib (n-1) + fib (n-2)

## 2 Canonical zipWith implementation

fib = 1 : 1 : zipWith (+) fib (tail fib)

## 3 With scanl

fib = fix ((1:) . scanl (+) 1)

## 4 With unfoldr

unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1)

## 5 A fairly fast version, using some identities

fib 0 = 0 fib 1 = 1 fib n | even n = f1 * (f1 + 2 * f2) | n `mod` 4 == 1 = (2 * f1 + f2) * (2 * f1 - f2) + 2 | otherwise = (2 * f1 + f2) * (2 * f1 - f2) - 2 where k = n `div` 2 f1 = fib k f2 = fib (k-1)

## 6 Another fast fib

fib = fst . fib2 -- | Return (fib n, fib (n + 1)) fib2 0 = (1, 1) fib2 1 = (1, 2) fib2 n | even n = (a*a + b*b, c*c - a*a) | otherwise = (c*c - a*a, b*b + c*c) where (a,b) = fib2 (n `div` 2 - 1) c = a + b

## 7 Fastest Fib in the West

This was contributed by wli

import System.Environment import Data.List fib n = snd . foldl fib' (1, 0) . map (toEnum . fromIntegral) $ unfoldl divs n where unfoldl f x = case f x of Nothing -> [] Just (u, v) -> unfoldl f v ++ [u] divs 0 = Nothing divs k = Just (uncurry (flip (,)) (k `divMod` 2)) fib' (f, g) p | p = (f*(f+2*g), f^2 + g^2) | otherwise = (f^2+g^2, g*(2*f-g)) main = getArgs >>= mapM_ (print . fib . read)

## 8 See also

Discussion at haskell cafe:

http://comments.gmane.org/gmane.comp.lang.haskell.cafe/19623