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This is one of my old ideas as a beginner as googologist. I never shared it before, but i will now share it.

NOTE: I don't remember well what all of the system was, but this is what i can remember. (i didn't archive the system, so i need to remember it from my memories)

The Graham Function

So, some people knows about the Graham Function, which is where my notation is based from. It is defined as follows:

G(0) = 4

G(n) = 3^...^3 w/ G(n-1) arrows

This is the function to create Graham's number, defined as G(64).

Grahams into Grahams

The next step into this system, is making the input of Graham Function be equal to G(64).

This way, we create G(G(64)), called Graham in Graham.

Similarly, we can repeat this process using Graham in Graham, to create G(G(G(64))), called Graham in Graham in Graham.

So, we have a pattern:

G(64), G(G(64)), G(G(G(64))),....

Each element uses the previous element as an input to the Graham Function.

Elements and sequences

So, let G(64) is first element, G(G(64)) is second element, G(G(G(64))) is third element, and G(n-1th element) is nth element.

This creates the first sequence. 

Now, how to get to the second sequence?

Easy.

First, take the G(64)th element of the first sequence.

Then, have a G(G(64))th element of the first sequence.

Then, have a G(G(G(64)))th element of the first sequence, and so on.

Another pattern forms:

G(64)th element of the first sequence, G(G(64))th element of the first sequence, G(G(G(64)))th element of the first sequence,...

The elements of this pattern forms the second sequence. Getting to the rest of sequences is also easy.

Just take the (nth element of the first sequence)th element of the mth sequence and this will be the nth element of the (m+1)th sequence.

This way, we can have multiple sequences.

Groups, gangs, and other group types

The sequences also form another pattern: a sequence contains the (elements of the first sequence)th elements of the previous sequence. And as such, we can also classify the elements of this pattern, as a group.

This can also be the first group, and similar to the sequences, we can have multiple groups.

But how?

Well, it's easy: The (nth element of the first sequence of the first group)th element of the (nth element of the first sequence of the first group)th sequence of the mth group is the nth element of the first sequence of the (m-1)th group.

The rest of elements from the rest of sequences can be calculated using the same method from the previous section, just in the mth group instead of the first group.

E.G: The nth element from the mth sequence from the xth group is the (nth element of the first sequence of the first group) from the (m-1)th sequence from the xth group.

Similarly, we can classify groups as gangs, E.G:

-=- WIP -=-

Untitled Notation, Part 1

Coming.

Untitled Notation, Part 2

Coming.

Untitled Notation, Part 3

Coming.

Untitled Notation, Part 4

Coming.

Untitled Notation, Part 5

Coming.

Untitled Notation, Part 6

Coming.

Untitled Notation, Part 7 (?)

Coming, maybe.

Multiple times

Coming.

Time types

Coming.

''Unnoficial'' Extensions

NOTE: The following are extensions that weren't originally in the system, due to my lack of remembering it.

Coming.

Attempt into making this into a notation

For better understanding, i will try to put this into a notation.

# is rest of expression.

& is any array.

G(n,m) = G(G(n,m-1))

G(n,1) = G(n)

G(#,1) = G(#)

-=- WIP -=-

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