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 \(a_{\rightarrow b}\).

2^^^^2 = 2^^^2 = 2^^2 = 2^2 = 4

3^^^^2 = 3^^^3 = tritri

4^^^^2 = 4^^^4 = 4^^4^^4^^4 = 4^^4^^4^4^4^4 = 4^^4^4^4^4^.......4^4^4^4 4's.......^4^4^4^4 = 4^4^4^4^....^4^4^4^4, with 4^4^4^4^....^4^4^4^4 4's, with 4^4^4^4 4's

5^^^^2 = 5^^^5 = 5^^5^^5^^5^^5 = 5^^5^^5^^5^5^5^5^5 = 5^^5^^5^5^5^.......5^5^5^5^5 5's.......^5^5^5 = 55 5 5 5 : : : : 5       with 55 5 5 5 : : : : 5        5's, with 55 5 5 5 : : : : 5        5's, with 55 5 5 5   5's

2^^^^3 = 2^^^2^^^2 = 2^^^2^^2 = 2^^^2^2 = 2^^^4 = 2^^2^^2^^2 = 2^^2^^2^2 = 2^^2^^4 = 2^^2^2^2^2 = 2^^2^2^4 = 2^^2^16 = 2^^65536 = 22 2 2 2 : : : : 2      , 65536 2's

3^^^^3 = 3^^^3^^^3 = 3^^^tritri = 3^^3^^3^^...^^3^^3^^3, tritri 3's, 33 3 3 3 : : : : 3      , with 33 3 3 3 : : : : 3        3's, with 33 3 3 3 : : : : 3        3's, with 33 3 3 3 : : : : 3        3's with .............. This is repeated tritri times!!!

You see a pattern? aa a a a : : : : a      , with aa a a a : : : : a        a's, with aa a a a : : : : a        a's, with aa a a a : : : : a        a's, with ................ This is repeated a^^^^(b-1) times!

a^^^^b = a^^^a^^^a...a^^^a^^^a (b a's) = {a,b,4}