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 the tritri, and 10 pentated to 100 is gaggol.

Sunir Shah uses the notation \(a * b\) to indicate this function. 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.

Examples
Here are some examples of pentation in action:

\(1 \uparrow\uparrow\uparrow b = 1\)

\(a \uparrow\uparrow\uparrow 1 = a\)

\(\uparrow\uparrow\uparrow 2 =\)

\(2 \uparrow\uparrow\uparrow\) \(= {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} =\) \(65,536\)

\(3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} =\)

Here are some larger examples:

\(\uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)

\(\uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)

\(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)

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