Hexation

Hexation or sextation refers to the 6th hyperoperation if addition is to be regarded as the first. It is equal to \(a \uparrow\uparrow\uparrow\uparrow b\) in Knuth's up-arrow notation. Sunir Shah uses the notation \(a **\ b\) to indicate this. Jonathan Bowers calls it 'a to the b'th layer'. Sbiis Saibian proposes the notation \(a_{\rightarrow b}\) in analogy to ba for tetration, but he usually uses up-arrows.

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.

Tim Urban calls hexation a "power tower feeding frenzy psycho festival".

Examples
Here are some examples of hexational-level calculations.

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

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

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

\(2 \uparrow^{4} 3 ={^{65,536}}2\) (a power tower of 2's 65,536 terms high)

\(3 \uparrow^{4} 2 = 3 \uparrow\uparrow\uparrow 3 = {^{7,625,597,484,987}}3\) (a power tower of 3's 7,625,597,484,987 terms high)

\(3 \uparrow^{4} 3 = 3 \uparrow\uparrow\uparrow \text{tritri} = g_{1}\)

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