Hexation

Hexation or sextation refers to the function \(a \uparrow\uparrow\uparrow\uparrow b\), where Arrow Notation is used. Sunir Shah uses the notation \(a **\ b\) to indicate this. Bowers calls it 'a to the b'th layer'. Sbiis Saibian uses the notation \(a_{\rightarrow b}\).

Hexation can be written in array notation as {a,b,4}, in chained arrow notation as \(a \rightarrow b \rightarrow 4\) and in Hyper-E notation as E(a)1#1#1#b.

Hexational growth rate is equivalent to \(f_5(n)\) in the fast-growing hierarchy.

Examples
Here are some examples of hexation in action:

\(1 \uparrow^{4} b = 1\)

\(a \uparrow^{4} 1 = a\)

\(2 \uparrow^{4} 2 = 4\)

\(2 \uparrow^{4} 3 =^{65536}2\)

\(\uparrow^{4} 2 = 3 \uparrow\uparrow\uparrow 3 = ^{7625597484987}3\)

\(3 \uparrow^{4} 3 = 3 \uparrow\uparrow\uparrow\)

\(4 \uparrow^{4} 2 = 4 \uparrow\uparrow\uparrow 4\) = \(^{^{^{4}4}4}4\)

E(10)1#1#1#b = 10↑↑↑↑b