Re: [Haskell-cafe] Course-of-value recursion by defining a sequence as a self-referential infinite list

19 Jan 2025


      Yes, the Fibonacci example and its connection to laziness is old! Not sure if it’s ever been named ‘course-of-value recursion’ or its corresponding strong induction.
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On Jan 19, 2025, at 10:15 AM, Henning Thielemann  wrote:
On Sun, 19 Jan 2025, Vanessa McHale wrote:
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Laziness turns out to allow course-of-value recursion where one might use memoization in other languages. But I hadn’t seen this articulated!
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Famously, one can use this to define the Fibonacci numbers, viz.
fibs :: [Integer]
fibs = 1 : 1: zipWith (+) fibs (tail fibs)
This example is old and well-known, right?
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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
I have:
catalanNumbers :: Num a => [a]
catalanNumbers =
  let xs = 1 : PowerSeries.mul xs xs
  in  xs
https://hackage.haskell.org/package/combinatorial-0.1.1/docs/src/Combinatori...