Step by Step solution about How to resolve the algorithm Hofstadter-Conway $10,000 sequence step by step in the Haskell programming language

The provided Haskell code is an implementation of Mallows's problem, which is about finding the maximum of a certain function hc and its derivative. The code also calculates Mallows's number, which is the smallest integer n such that hc(n) < 0.55.

Let's go through the code step by step:

• Data Structures:
  • The code uses several data structures from the Haskell standard library:
    • Data.List: Provides functions for working with lists.
    • Data.Ord: Provides ordering functions.
    • Data.Array: Provides functions for working with arrays.
    • Text.Printf: Provides functions for formatted printing.
• hc Function:
  • The hc function takes an integer n as input and returns an Array of integers. The Array is constructed using the listArray function, which creates an array from a list.
  • The list used to initialize the array is constructed as follows:
    • The first element is 1.
    • The second element is 1.
    • For i from 3 to n, the i-th element is calculated using the f function applied to the previous element of the array and i.
• f Function:
  • The f function takes two integers a and i as input and returns an integer.
  • It calculates the value at i in the hc array using the value at i - 1 and the value at i - a(i - 1).
• printMaxima Function:
  • The printMaxima function takes a tuple (n, (pos, m)) as input and prints the maximum value m of hc between 2^n and 2^(n + 1) at position pos.
• main Function:
  • The main function is the entry point of the program.
  • It calculates the following:
    • hca: The hc array with a length of 2^20.
    • hc': A function that takes an integer n and returns the value of hc(n) divided by n.
    • maxima: A list of tuples (pos, m) where pos is the position of the maximum value and m is the maximum value of hc between 2^n and 2^(n + 1) for n from 0 to 19.
    • max: A function that takes a list of integers seq and returns the maximum value and its position in seq.
    • powers: A list of lists of integers where each sublist contains the powers of 2 from 2^n to 2^(n + 1) - 1 for n from 0 to 19.
    • mallows: The smallest integer n such that hc(n) < 0.55.
  • Finally, the program prints the maxima and Mallows's number.

import Data.List
import Data.Ord
import Data.Array
import Text.Printf

hc :: Int -> Array Int Int
hc n = arr
  where arr = listArray (1, n) $ 1 : 1 : map (f (arr!)) [3 .. n]
        f a i = a (a $ i - 1) + a (i - a (i - 1))

printMaxima :: (Int, (Int, Double)) -> IO ()
printMaxima (n, (pos, m)) =
    printf "Max between 2^%-2d and 2^%-2d is %1.5f at n = %6d\n"
                             n    (n + 1)        m        pos

main = do
    mapM_ printMaxima maxima
    printf "Mallows's number is %d\n" mallows
  where
    hca = hc $ 2^20
    hc' n  = fromIntegral (hca!n) / fromIntegral n
    maxima = zip [0..] $ map max powers
    max seq = maximumBy (comparing snd) $ zip seq (map hc' seq)
    powers = map (\n -> [2^n .. 2^(n + 1) - 1]) [0 .. 19]
    mallows = last.takeWhile ((< 0.55) . hc') $ [2^20, 2^20 - 1 .. 1]