Hexation

Hexation or sextation refers to the 6th hyperoperation if addition is to be regarded as the first. It is equal to the binary function \(\uparrow\uparrow\uparrow\uparrow \) in Knuth's up-arrow notation.

"a hexated to b" 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. 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→b in analogy to ba for tetration, but he usually uses up-arrows.

Hexational growth rate is approximately \(f_5(n)\) 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↑↑↑↑b=1

a↑↑↑↑1=a

2↑↑↑↑2=4

2↑↑↑↑3 =2↑↑65,536 (a power tower of 2's 65,536 terms high)

3↑↑↑↑2 = 3↑↑↑3 = 3↑↑7,625,597,484,987 = tritri (a power tower of 3's 7,625,597,484,987 terms high)

3↑↑↑↑3 = 3↑↑↑tritri = g1 = grahal

4↑↑↑↑2 = 4↑↑↑4 = \(^{^{^{4}4}4}\! 4\) where \({^b} a\) denotes tetration

3 ↑↑↑↑ 8 = 3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3

Appearance in other googolisms
Hexation is used in \(G_1\), the first value of the Graham function, which is equal to 3 hexated to 3. This number is also known as grahal.