Hexation

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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'. Reuben Goodstein mentions that \(a_{b}\) would stand for pentation, analogous to how \(^{b}a\) would stand for tetration, as well as commenting that such superscripts/subscripts cannot be used for higher hyperoperators; similarly Sbiis Saibian proposes the notation a→_b in analogy to ^ba for tetration, but he usually uses up-arrows.

In hyperlicious function, it is equal to \(h_4(a,b)\).

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.

Pseudocode

Below is an example of pseudocode for hexation.

function hexation(a, b):
    result := 1
    repeat b times:
        result := a pentated to result
    return result

See also

• Tritri

Hyper operators

Main article: hyper operator

Hyper-operators: successor · addition · multiplication · exponentiation · tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulation
Bowers' extensions: expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation
Saibian's extensions: hexonation · heptonation · octonation · ennonation · deconation
Tiaokhiao's extensions: megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...