User blog:DJGrex/Super Staging Factorial Function

Let me introduce the function based on factorial (!). I will call this function Super Staging Factorial Function. In this function we can go far beyond with the simple notation !. I gonna tell how I got this idea. I mainly got this inspiration by dysfunctionalfactorial's idea at http://www.artofproblemsolving.com/community/c1664h1003843 about factorial. This is not be confused with Faxul terms. The naming system focuses with the number 200, similar to Faxul terms. I use Faxul terms to see how fast this function grow perspective. I don't think this function would beat Expofaxul terms very fast. Numbers I have created is based on the name Lexol, meaning "Level Factorial", the first tier. And name Raxol, meaning "Rank Factorial", the second tier. I hasn't gotten to the third yet. We'll use pairs of vertical pipes (|) with number or chain (\alpha) within those pairs of pipes, and a number in the subscript (\beta) outside the pipes, (AND if necessery, have a number superscript (\gamma) which describes the repeat of the function), and aswell as a input (x) to make those process:


 * \alpha|\beta\gamma(x)=output

Let's clearify some rules:


 * Rule 1: numbers in this function must be 1 or greater
 * Rule 2: |1|1(x)=x!
 * Rule 3: |m|n1(x)=|m|n(x)
 * Rule 4: |m|no+1(x)=|m|no(|m|n1(x))
 * Rule 5: |m|n+11(x)=|m|nx(x)
 * Rule 6: |m+1|1(x)=|m|x(x)
 * Rule 7: |...|1|...(n+1 pairs of |'s)...|1(x) = |...|x|...(n pairs of |'s)...|1(x)
 * Rule 1,3-6 is also applied on multiple |'s
 * Rule 8: |...|m|...(n pairs of |'s)...|undefined(x) = |n,m|1(x)
 * Rule 9: |m+1,11,12,...,1n+1|1(x)=|m,x,11,12,...,1n|1(x)

Examples:


 * 1|1(3)=3!=3·2·1=6
 * 1|1(10)=10!=3.628.800
 * 1|1(100)=100!≈9,33262154439·10157
 * 1|1(200)=200!≈7,9·10375, I name this number Lexol. Also equal to Faxul coined by
 * 1|1(|1|1(3))=(3!)!=720, this can be compressed into |1|12(3).
 * 1|12(10)=(10!)!≈5,2279·1022228109
 * 1|12(200)=(200!)!≈109,48539·10 378, which is named Kilolexol

More names:
 * Megalexol
 * Gigalexol
 * Dialexol
 * Trialexol
 * Grand Lexol
 * Bigrand Lexol
 * Ultimate Lexol
 * Uldimate Lexol
 * Ulquintimega Quadgrand Exatetralexol

with Extended Notation:
 * Raxol
 * Ultimate Raxol

I apologizes if a spelled incorrectly or "the red line" isn't really good.

