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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?

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    $\begingroup$ That is an incredibly fast growing function...Letting $f(1)=1$ and $f(n)=n^{f(n-1)}$ we have already $f(4)$ is on the order of $10^{106000}$ and $f(5)$ obscenely large beyond even that. $\endgroup$ – JMoravitz May 26 '16 at 19:12
  • $\begingroup$ Tower of Babel? $\endgroup$ – John Wayland Bales May 26 '16 at 19:14
  • $\begingroup$ Sadly exponentiation loses a lot of properties which summation and multiplication have in analogy. However yours is an interesting concept, unfortunately exponentiation and even less tetration and so on are not studied much as sum and multiplication because of that loss of properties. $\endgroup$ – Renato Faraone May 26 '16 at 19:57
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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.

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