Laziness turns out to allow course-of-value recursion where one might use memoization in other languages. But I hadn’t seen this articulated!
Famously, one can use this to define the Fibonacci numbers, viz.
fibs :: [Integer]
fibs = 1 : 1: zipWith (+) fibs (tail fibs)
Or the Catalan numbers:
catalan :: [Integer]
catalan = 1 : 1 : [ sum [ (-1)^(k+1) * (pc (n - ((k*(3*k-1)) /. 2)) + pc (n - ((k*(3*k+1))/.2))) | k <- [1..n] ] | n <- [2..] ]
where
pc m | m >= 0 = part !! m | otherwise = 0
infixl 6 /.
(/.) = quot
Cheers,
Vanessa McHale