The Fibonacci sequence

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

Naive solution

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

Canonical zipWith implementation

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

With scanl

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

With unfoldr

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

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)

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

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)

See also

• Discussion at haskell cafe
• Some other nice solutions