C function

C function is a function defined by SeveralLegend9998.SeveralLegend9998's New Googology Series (Retrieved 3nd of June, 2024)(SeveralLegend9998's Googology Series (Retrieved 23rd of October, 2022) for former definition)
==Linear C Function==
The rules of Linear C Function goes as follows: 
#With one entry after C, aCb = ab where a and b are positive integers
#Last entry is 1, aC#/1 = aC# where # denotes the rest of the expression
#First entry after C is greater than 1, aCb/# = (ab)C1#
#The entry that comes before C is 1, 1C# = 1
#Otherwise, aC$/1/b/# = aC$2/((a-1)C$/1/b/#)/(b-1)/# where $ is a bunch of slash-separated 1's, and $2 is a bunch of slash-separated a's
===Examples===
3C1/3 = 3C(2C1/3)/2
= 3C(2C(1C1/3)/2)/2
= 3C(2C1/2)/2
= 3C(2C(1C1/2)/1)/2
= 3C(2C(1C1/2))/2 
= 3C(2C1)/2 
= 3C2/2 
= (3*2)C1/2 
= 6C1/2 
= 6C(5C1/2) 
= ... 
= 6C(5C(4C(3C(2C(1C1/2)))))
= 6C(5C(4C(3C(2C1)))) 
= 6C(5C(4C(3C2))) 
= ... 
= 720 
= 6!
Tip: nC1/2 = n!
4C1/3 
= 4C(3C1/3)/2 
= ... (See the example above) ... 
= 4C720/2 
= 2880C1/2 
= ... (See the tip above) ... 
= 2880!
5C1/3 
= 5C(4C1/3)/2 
= ... (See the example above) ... 
= 5C(2880!)/2 
= (2880!*5)C1/2 
= ... (See the tip above) 
= (2880!*5)!
3C1/1/2 
= 3C3/(2C1/1/2)/1 
= 3C3/(2C2/(1C1/1/2)/1)/1 
= 3C3/(2C2/(1C1/1/2)/1) 
= 3C3/(2C2/(1C1/1/2)) 
= 3C3/(2C2/(1C1/1/2)) 
= 3C3/(2C2/1) 
= 3C3/(2C2) 
= 3C3/4
= 9C1/4 
= ...
===Comparison with the Fast-growing hierarchy (Intended) ===
aCa ≈ \(f_1(a)\)
aC1/2 ≈ \(f_2(a)\)
aCa/2 ≈ \(f_2(f_1(a))\)
aC(aCa/2)/2 ≈ \(f_2(f_2(f_1(a)))\)
aC1/3 ≈ \(f_3(a)\)
aC1/4 ≈ \(f_4(a)\)
aC1/a ≈ \(f_ω(a)\)
aC1/1/2 ≈ \(f_{ω+1}(a)\)
aC1/2/2 ≈ \(f_{ω+2}(a)\)
aC1/a/2 ≈ \(f_{ω2}(a)\)
aC1/1/3 ≈ \(f_{ω2+1}(a)\)
aC1/a/a ≈ \(f_{ω^2}(a)\)
aC1/1/1/2 ≈ \(f_{ω^2+1}(a)\)
aC1/a/1/2 ≈ \(f_{ω^2+ω}(a)\)
aC1/a/2/2 ≈ \(f_{ω^2+ω2}(a)\)
aC1/a/a/2 ≈ \(f_{ω^2*2}(a)\)
aC1/a/a/3 ≈ \(f_{ω^2*3}(a)\)
aC1/a/a/a ≈ \(f_{ω^3}(a)\)
aC1/a/a/a/a ≈ \(f_{ω^4}(a)\)
aC1/a/a/a/.../a/a/a ≈ \(f_{ω^ω}(a)\) (The intended limit of linear C function)
==Extened C Function==
Extended C Function uses multiple slashes between entries, and works like this:
#Without any slashes, aCb = a*b
#Last entry is 1: aC#1 = aC#
#First entry after C is greater than 1: aCb/# = (a*b)C1#
#The entry that comes before C is 1: 1C# = 1
#If x < y: #(/^x)1(/^y)#2 = #(/^y)#2 where /^k = ///.../// with k /'s, and x and y are positive integers
#First >1 entry comes right after multiple slashes: aC$(/^x)b# = aC$(/^(x-1))1(/^(x-1))1...1(/^(x-1))1(/^(x-1))2(/^x)b-1# with a (/^(x-1))'s where x > 1
#Otherwise: aC$/1/b/# = aC$2/((a-1)C$/1/b/#)/(b-1)/#
===Examples===
3C1//2
= 3C1/1/1/2//1
= 3C1/1/1/2
= 3C3/3/(2C1/1/1/2)/1
= 3C3/3/(2C1/1/1/2)
= 3C3/3/(2C2/2/(1C1/1/1/2)/1)
= 3C3/3/(2C2/2/1/1)
= 3C3/3/(2C2/2)
= 3C3/3/(4C1/2)
= 3C3/3/4!
= 3C3/3/24
= ...
3C1///2////4
= 3C1//1//1//2///1////4
= 3C1//1//1//2////4
= 3C1//1//1/1/1/2//1////4
= 3C1//1//1/1/1/2////4
= ...
===Comparison with FGH (Intended)===
nC1//2 ≈ \(f_{\omega^\omega}(n)\)
nCn//2 ≈ \(f_{\omega^\omega}(f_1(n))\)
nC1/2//2 ≈ \(f_{\omega^\omega+1}(n)\)
nC1/n//2 ≈ \(f_{\omega^\omega+\omega}(n)\)
nC1/n/n//2 ≈ \(f_{\omega^\omega+\omega^2}(n)\)
nC1//3 ≈ \(f_{\omega^\omega2}(n)\)
nC1//n ≈ \(f_{\omega^{\omega+1}}(n)\)
nC1//1/2 ≈ \(f_{\omega^{\omega+1}+1}(n)\)
nC1//n/2 ≈ \(f_{\omega^{\omega+1}2}(n)\)
nC1//n/n ≈ \(f_{\omega^{\omega+2}}(n)\)
nC1//1//2 ≈ \(f_{\omega^{\omega2}}(n)\)
nC1///2 ≈ \(f_{\omega^{\omega^2}}(n)\)
nC1////2 ≈ \(f_{\omega^{\omega^3}}(n)\)
nC1/////.../////2 ≈ \(f_{\omega^{\omega^\omega}}(n)\) (Limit of this extension)
==Sources==
Category:Functions
Category:Notations
Category:Unformalised