Step by Step solution about How to resolve the algorithm Mutual recursion step by step in the Mathematica/Wolfram Language programming language

This code defines a pair of recursive functions called f and m that are used to generate two sequences of numbers. The sequences are defined as follows:

f[n] = n - m[f[n-1]]
m[n] = n - f[m[n-1]]

The code first defines base cases for the functions f and m when n = 0. Then, it defines the recursive definitions of the functions.

f[0] := 1
m[0] := 0

f[n_] := n - m[f[n - 1]]
m[n_] := n - f[m[n - 1]]

The code then uses the /@ operator to apply the functions f and m to the range of integers from 0 to 29. The results are printed to the console, showing the values of f[n] and m[n] for each value of n in the range.

The output is as follows:

{0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19}
{1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19}

The sequence generated by f[n] is the Fibonacci sequence, which is a sequence of numbers where each number is the sum of the two preceding numbers. The sequence generated by m[n] is the Lucas sequence, which is a similar sequence to the Fibonacci sequence, but with different starting values.

f[0]:=1
m[0]:=0
f[n_]:=n-m[f[n-1]]
m[n_]:=n-f[m[n-1]]


f[0]:=1
m[0]:=0
f[n_]:=f[n]=n-m[f[n-1]]
m[n_]:=m[n]=n-f[m[n-1]]


m /@ Range[30] 
f /@ Range[30]


{0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,17,18,19}
{1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12,13,13,14,14,15,16,16,17,17,18,19}