Pentation

Pentation or quintation refers to the function \(a \uparrow\uparrow\uparrow b\), where Arrow Notation is used. It produces numbers very much larger than those produced by tetration.

Pentation can be written in array notation as {a,b,3}, in chained arrow notation as \(a \rightarrow b \rightarrow 3\) and in Hyper-E notation as E(a)1#1#b.

Pentation is less known than its tetrational cousin, but there are a few googologisms employing it: 3 pentated to 3 is known as tritri, and 10 pentated to 100 is gaggol.

Sunir Shah uses the notation \(a * b\) to indicate this function. Jonathan Bowers calls it "a to the b'th tower". Sbiis Saibian uses the notation \(_{b \leftarrow}a\).

Pentational growth rate is equivalent to \(f_4(n)\) in the fast-growing hierarchy.

A strip from the webcomic ' suggested the name "penetration'''" in humorous analogy with sexation.

Examples
Here are some small examples of pentation in action:


 * \(1 \uparrow\uparrow\uparrow b = 1\)
 * \(a \uparrow\uparrow\uparrow 1 = a\)
 * \(2 \uparrow\uparrow\uparrow 2 = 4\)
 * \(2 \uparrow\uparrow\uparrow 3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} = 65,536\)
 * \(3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} = 7,625,597,484,987\)

Here are some larger examples:


 * \(3 \uparrow\uparrow\uparrow 3 = {^{^{3}3}3} = {^{7,625,597,484,987}3}\) = tritri, a power tower of 7,625,597,484,987 threes
 * \(5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)
 * \(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)
 * \(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)

Pseudocode
Below is an example of pseudocode for pentation.

function pentation(a, b): result := 1 repeat b times: result := a tetrated to result return result