Exponent analog to the factorial function

Question

Triangular numbers can be discovered by taking any number $n$, and adding

$$\sum_{i=0}^n i = n + (n - 1) + (n - 2) ... 1 = \frac{n(n + 1)}{2}$$

These numbers can be generalized by putting any real argument in the expression above. Factorial is the function which is equivalent to multiplying

$$n(n - 1)(n - 2)...2*1$$

for whole numbers, and it can be generalized with the gamma function. Is there an a name for the exponent analog to the factorial function:

$$f(n) = n^{(n - 1)^{(n - 2)^{.^{.^{.^{1}}}}}}$$

and if so, is there a formula for a smooth curve that passes through the points for the natural arguments and generalizes it for the reals?

Answer 1

Check out tetration: https://en.wikipedia.org/wiki/Tetration#Iterated_powers_vs._iterated_exponentials

The actual function you are describing is on that page as well, and is called a nested exponential. Tetration is a special case of a nested exponential. For the record, pentation is the next operation up (it's tetration nested like tetration is exponentiation nested), then hexation and so forth.