User blog:ArtismScrub/Generalized Bowerian G Function Notation

See G function.

Definition
Gn[a] = Ga using base n G function

Gn[a,b] = Gba, or GGGG...(b "G"s)...GGGGa using base n G function

G n [a,b,c] = (G{c}b)a, or (G ↑↑↑↑...(c arrows)...↑↑↑↑b)a using base n G function

a always comes at the end of an expression!

Examples
G 3 [3] = G3 in base 3 = 3 ↑ ↑ ↑3 = Tritri

G 3 [4,64] =  G 64 4 in base 3 = Graham's number

G 3 [1,3,2] = (3G) = G GG =  G G3 =  G GGG =  G GG3  =  G G(tritri)  =  G tritriplex  = 3{3{3{3{...{3{3{3{3}3}3}3}...}3}3}3 with 3{3{3}3}3 "3"s from the center to either end outwards = 3 multiexpanded to 3

G n [1,b,c]  = {n,b,c,2} in BEAF

We can generalize this to something like:  G n [a,b,c,d] = ({G,b,c,d})a, and continue through array notation, but {a,b,1,3} and beyond quickly dominates the growth rate of the G function, and we want to keep the function useful. So...

Higher order G functions
We can define further extension to the G function.

What about a G in base G?

G n,1 [a,b,c] =  G n [a,b,c]

G n,m [a,b,c] =  G <sub style="font-weight:400;">Gn,m-1 [a,b,c]

Examples
G <sub style="font-weight:400;">4,2 [4] = G4 in base base-4 G = G ↑↑↑↑G in base 4 =  G <sub style="font-weight:400;">4 [1,4,4] in first-order G notation = {4,4,4,2} in BEAF

G <sub style="font-weight:400;">4,2 [1,4] = GGGG in base base-4 G = G{G{G{G}G}G}G in base 4 =  G <sub style="font-weight:400;">4 [1,4,1,2] in first-order G notation = {4,4,1,3} in BEAF

G <sub style="font-weight:400;">n,m [1,b,c]  = {n,b,c,m+1} in BEAF

How to continue this??
So, I've basically created an alternative, more complex method of expressing what can easily be expressed using tetrentrical arrays, reaching growth rate  ω2 at its limits.

Well... so what? Like I just said, these expressions can be created in BEAF much easier.

Plus, I can't think of how to continue this... How can I make this reach into pentetrical range and beyond without diverging from BEAF? What would  G <sub style="font-weight:400;">n,m,o [a,b,c] solve to? I have no idea.

For now, I'll just leave it at this, I suppose.