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

This Wolfram code explores the Mallows number sequence, a sequence of integers where each term is defined recursively as the sum of the previous term and the term before that.

1. Initialization: The first two terms of the sequence, a[1] and a[2], are defined as 1.

2. Recursive Definition: For any integer n greater than 2, the term a[n] is calculated as the sum of a[a[n-1]] and a[n-a[n-1]]. This recursive definition creates a sequence where each term depends on the previous terms in a complex way.

3. Looping and Printing: The code uses the Map function to iterate over a range of values from 1 to 19 and print the maximum value of the sequence a[n]/n for each power of 2 in that range. The Table function is used to generate the values of a[n]/n for different values of n within each power of 2 range.

4. Finding the Mallows Number: The code also finds the "Mallows number," which is the smallest value of n such that a[n]/n is less than or equal to 0.55. This is done using a While loop that decrements the value of n until the condition is met.

The output of the code will show the maximum values of a[n]/n for different powers of 2, as well as the Mallows number, which is approximately 2^20 or 1,048,576.

a[1] := 1; a[2] := 1;
a[n_] := a[n] = a[a[n-1]] + a[n-a[n-1]]

Map[Print["Max value: ",Max[Table[a[n]/n//N,{n,2^#,2^(#+1)}]]," for n between 2^",#," and 2^",(#+1)]& , Range[19]]
n=2^20; While[(a[n]/n//N)<0.55,n--]; Print["Mallows number: ",n]