Hyperlicious function

The hyperlicious function is defined as
*\(h_x(a,b) = a\uparrow^{x-2} b \),
*\(h_x(a,b,c,\ldots,m,n,1) = h_x(a,b,c,\ldots,m,n)\), and
*\(h_x(a,b,c,\ldots,m,n) = h_{h_x(a,b,c,\ldots,m - 1)}(a,b,c,\ldots,m)\).A googol is a tiny dot
==Up to \(h_x(a,b)\)==
* \(h_0(a,b) = a+1\) (successor function)
* \(h_1(a,b) = a+b\) (addition)
* \(h_2(a,b) = ab\) (multiplication)
* \(h_3(a,b) = a^b\) (exponentiation)
* \(h_4(a,b) = a \uparrow\uparrow b\) (tetration)
* \(h_5(a,b) = a \uparrow\uparrow\uparrow b\) (pentation)
* \(h_6(a,b) = a \uparrow\uparrow\uparrow\uparrow b\) (hexation)
* \(h_7(a,b) = a \uparrow\uparrow\uparrow\uparrow\uparrow b\) (heptation)
* \(h_8(a,b) = a \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow b\) (octation)
* \(h_9(a,b) = a \uparrow^7 b\) (enneation)
* \(h_{10}(a,b) = a \uparrow^8 b\) (decation)
* \(h_{20}(a,b) = a \uparrow^{18} b\) (vigintation)
* \(h_{100}(a,b) = a \uparrow^{98} b\) (centation)
==Examples==
*\(h_3(2,6) = 2 \uparrow 6\)
*\(h_3(10,100) = 10 \uparrow 100\) (googol)
*\(h_4(10,100) = 10 \uparrow\uparrow 100\) (giggol)
*\(h_5(10,100) = 10 \uparrow\uparrow\uparrow 100\) (gaggol)
*\(h_4(2,6,2) = h_{h_4(2,5)}(2,6) = h_{2\uparrow^6 5}(2,6) = 2 \uparrow^{2 \uparrow^{6} 5-2} 6\)
==Growth rate==
\(h(x) = h_x(x,x...x,x)\) with x x's in the array is approximately \(f_{\omega+1}(x)\) in the fast growing hierarchy, which is closely related to the expansion function ({x,x,1,2}). The two-entry hyperlicious function (\(h_x(a,b)\)) eventually dominates any hyper-operator, such as tetration, pentation, or even beyond centation.
== Sources ==
== See also ==
* ultron
* wakoogol
* wakoogolplex
ja:ハイパーリシャス関数
Category:Notations
Category:Functions