Let be a function defined by the following:
being and natural numbers.
What is the value of ?
Let’s start with the values of :
Continuing the process we get
ending with the expression
Now we can try to get the values of :
Repeating the process we end with
Using this result to get :
Iterating this process we reach
is finally in sight 😀 Replacing:
Making some additional iterations to get the pattern:
Now the progression is clear: each iteration adds one 2 to the exponential tower and the term moves with . Let’s define a function to encapsulate the tower exponential:
( corresponds to a tower of 2s).
Then we can iterate until reaching
Then the asked for value is in our notation.