Step by Step solution about How to resolve the algorithm Thue-Morse step by step in the Julia programming language

Function Definition:

function thuemorse(len::Int)

This defines a function called thuemorse that takes one argument, len, which specifies the length of the Thue-Morse sequence to generate.

Initialization:

   rst = Vector{Int8}(len)
   rst[1] = 0

• rst is initialized as a growing Vector{Int8} of length len.
• Its first element rst[1] is set to 0, as per the definition of the Thue-Morse sequence.

Loop for Generating the Sequence:

   i, imax = 2, 1
   while i ≤ len
       while i ≤ len && i ≤ 2 * imax
           rst[i] = 1 - rst[i-imax]
           i += 1
       end
       imax *= 2
   end

• i is the index of the element in the sequence being calculated.
• imax keeps track of the current maximum value of i that has been reached in previous iterations.
• The inner loop generates the elements of the sequence using the Thue-Morse definition:
  • If i is less than or equal to the length len and less than or equal to twice imax, then:
    • The value of rst[i] is set to 1 minus the value of rst[i-imax]. This is the definition of the Thue-Morse sequence, where each element is the complement (0 or 1) of the element imax steps back in the sequence.
    • i is incremented by 1.
• The outer loop continues doubling imax after each iteration of the inner loop. This allows the sequence to be generated efficiently by doubling the steps back with each iteration.

Return:

   return rst

The function returns the generated Thue-Morse sequence as a Vector{Int8}.

Usage:

println(join(thuemorse(100)))

• This line generates the Thue-Morse sequence of length 100 and joins its elements into a string using join.
• println then prints the resulting string.

function thuemorse(len::Int)
    rst = Vector{Int8}(len)
    rst[1] = 0
    i, imax = 2, 1
    while i ≤ len
        while i ≤ len && i ≤ 2 * imax
            rst[i] = 1 - rst[i-imax]
            i += 1
        end
        imax *= 2
    end
    return rst
end

println(join(thuemorse(100)))