User blog:Pteriforever/Pteriforever's W-function

This is my first serious attempt at googology. I hope I didn't screw up too badly ^.^

The Basic Idea of My W-function
The W-function takes the following form:

\[W(X_n,X_{n-1},...,X_2,X_1\]

Rule 1. If there is only one entry:

\[W(X) = 2 ^ X\]

Rule 2. If there are two or more entries, with no leading zeroes:

\[W(X_n,X_{n-1},...,X_1,X_0) = W(X_n-1,\underbrace{W(X_n-1,X_{n-1},...,X_1,X_0),...,W(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

This is less confusing when looking at the 2-entry instance:

\[W(X_2,X_1) = W(X_2-1,W(X_2-1,X_1))\]

Rule 3. Leading zeroes are deleted if they exist:

\[W(0, X_n,X_{n-1},...,X_1) = W(X_n,X_{n-1},...,X_1)\]

First Extension: WW-function
Let

\[WW(X) = W(\underbrace{X,X,X...,X,X,X}_X)\]

\[WW^n(X) = \underbrace{WW(WW(WW...(WW(X))...))}_n\]

Second Extension: WW[n]Y-Function
Let

\[WW[1]Y = WW^{WW(X)}(X)\]

\[WW[n]Y = WW[n-1]Y^{WW[n-1]Y(X)}(X)\]

One-entry level
\[W(0) = 2^0 = 1\]

\[W(1) = 2^1 = 2\]

\[W(2) = 2^2 = 4\]

\[W(3) = 2^3 = 8\]

\[W(5) = 2^5 = 32\]

\[W(10) = 2^10 = 1024\]

\[W(16) = 2^16 = 65536\]

\[W(20) = 2^20 = 1048576\]

Two-entry level
\[W(1,0) = 2\]

\[W(1,1) = 4 = 2^2\]

\[W(1,2) = 16 = 2^{2^2}\]

\[W(2,0) = 16 = 2^{2^2}\]

\[W(1,3) = 256 = 2^{2^3}\]

\[W(1,4) = 65536 = 2^{2^{2^2}}\]

\[W(2,1) = 65536 = 2^{2^{2^2}}\]

\[W(2,2) = 2^{2^{2^3}}\]

\[W(3,0) = ^72\]

\[W(4,0) = ^{15}2\]

Multiple-entry level
\[W(1,0,0) = 4\]

\[W(1,0,1) = 16\]

\[W(1,1,0) = 2^{2^{2^3}}\]

\[W(1,0,2) = ^{18}2 = \mbox{Aecronul}\]

Multiple-entry level
\[W(1,1,1) = ^{18}2 = \mbox{Aecronul}\]

\[W(3,0,0) = \mbox{Aecraxul}\]

\[W(3,3,3) = \mbox{Aecronaxul}\]

\[W(1,1,1,1) = \mbox{Erlfuronul}\]

\[W(4,0,0,0) = \mbox{Erlfuraxul}\]

\[W(4,4,4,4) = \mbox{Erlfuronaxul}\]

\[W(1,1,1,1,1) = \mbox{Emtwonul}\]

\[W(5,0,0,0,0) = \mbox{Emtwaxul}\]

\[W(5,5,5,5,5) = \mbox{Emtwonaxul}\]

WW level
\[WW(5) = \mbox{Emtwonaxul}\]

\[WW(6) = \mbox{Musthonaxul}\]

\[WW(7) = \mbox{Ouvcosonaxul}\]

\[WW(8) = \mbox{Exronaxul}\]

WW level
\[WW(8) = \mbox{Exronaxul}}\]

\[WW(9) = \mbox{Faezonaxul}}\]

\[WW(10) = \mbox{Seritonaxul}}\]

\[WW(100) = \mbox{Googonaxul}}\]

\[WW(WW(8)) = \mbox{Drexronaxul}\]

\[WW(WW(WW(8))) = \mbox{Trexronaxul}\]

\[WW(WW(WW(WW(8)))) = \mbox{Tetrexronaxul}\]

Iterated WW level
\[WW^5(8) = \mbox{Pentexronaxul}\]

\[WW^6(8) = \mbox{Hextexronaxul}\]

\[WW^7(8) = \mbox{Heptexronaxul}\]

\[WW^8(8) = \mbox{Octexronaxul}\]

\[WW^9(8) = \mbox{Ennexronaxul}\]

\[WW^WW(8)(8) = \mbox{Exronaxulplex}\]

WW[n]Y level
\[WW[1]Y(8) = \mbox{Exronaxulplex}\]

\[WW[1]Y(WW(8)) = \mbox{Drexronaxulplex}\]

\[WW[1]Y(WW^2(8)) = \mbox{Trexronaxulplex}\]

\[WW[2]Y(8) = \mbox{Exronaxuldrex}\]

\[WW[2]Y(WW(8)) = \mbox{Drexronaxuldrex}\]

\[WW[3]Y(8) = \mbox{Exronaxultrex}\]

\[WW[4]Y(8) = \mbox{Exronaxultetrex}\]

\[WW[5]Y(8) = \mbox{Exronaxulpentex}\]

\[WW[6]Y(8) = \mbox{Exronaxulhextex}\]

\[WW[7]Y(8) = \mbox{Exronaxulheptex}\]

\[WW[8]Y(Y) = \mbox{Exronaxuloctex}\]

\[WW[9]Y(Y) = \mbox{Exronaxulennex}\]

\[WW[WW(8)]Y(8) = \mbox{Dypexronaxul}\]

\[WW[WW^2(8))]Y(8) = \mbox{Dydrexronaxul}\]

\[WW[WW[1]Y(8)]Y(8) = \mbox{Dypexronaxulplex}\]