Question

How can i write a program that computes f[n] (for Fibonacci numbers:f[n]=f[n]-f[n-2], with f[0] = any number) using Module and a While loop?

Answer 1

Homework? I hope you learn by example. 😉

Your subject line says recursion, but you don’t specify that in your question; rather, you specify Module and While. I’ll go with the latter.

fib[n_] := 
  Module[{x = 1, y = 0, i = 0},
    While[i++ < n, {x, y} = {y, x + y}];
    y
  ]

Array[fib, 7]

(*  Out[]= {1, 1, 2, 3, 5, 8, 13}  *)

Table[fib[m], {m, 1,10}] 

(*  Out[]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}  *)