Pentation

Pentation refers to the 5th hyperoperation starting from addition. It is equal to \(a \uparrow\uparrow\uparrow b\) in Knuth's up-arrow notation and since it is repeated tetration, it produces numbers that are much larger. Just a simple \(2 \uparrow\uparrow\uparrow 3\) give an amazing 65,536.
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 tetration, 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". Reuben Goodstein mentions that \(_{b}a\) would stand for pentation, analogous to how \(^{b}a\) would stand for tetration; likewise Sbiis Saibian proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.
Pentational growth rate is comparable to \(f_4(n)\) in the fast-growing hierarchy.
A strip from the webcomic suggested the name "penetration" in humorous analogy with sexation.http://www.smbc-comics.com/?id=2615
Tim Urban calls pentation a "power tower feeding frenzy".Prömel, H. J.; Thumser, W.; Voigt, B. "Fast growing functions based on Ramsey theorems", Discrete Mathematics, v.95 n.1-3, p. 341-358, Dec. 1991 .From 1,000,000 to Graham’s Number. Wait But Why.
In Notation Array Notation, it is written as (a{3,3}b).
In hyperlicious function, it is equal to \(h_5(a,b)\).
Graham, Rothschild and Spencer call the function \(f_4(n)\) in the fast-growing hierarchy, which is faster than \(2\uparrow\uparrow\uparrow n\) the WOW function, and corresponding growth rate wowzer.R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, 2nd edition
== Examples ==
Here are some small examples of pentation in action:
*\(1 \uparrow\uparrow\uparrow b = 1\)
*\(a \uparrow\uparrow\uparrow 1 = a\)
*\(2 \uparrow\uparrow\uparrow 2 = 4\)
*\(2 \uparrow\uparrow\uparrow 3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} = 65,536\)
*\(3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} =\) 
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
*\(5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)
*\(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)
*\(4 \uparrow\uparrow\uparrow 4 = {^{^{^{4}4}4}4}\)
*\(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)
== 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
== Sources ==
== See also ==
ja:ペンテーション
Category:Functions
Category:Binary operators
Category:Hyper operators