User blog:King2218/BAL Notation Part 1

This notation is based on the Steinhaus-Moser Notation. For now let's call this BA and not BAL. (BASE, ARRAY, LEVEL; no LEVEL for now :P )

Definition
Let $$\#$$ denote rest of array. Let $$a$$ and $$b$$ be natural numbers.

Rules: Obviously, the ARRAY is inside the curly braces.
 * $$a\{0\} = a$$ (a is called the BASE)
 * $$a\{\#, 0\} = a\{\#\}$$ (I chose 0 for some reason)
 * $$a\{\underbrace{0, 0, \cdots, 0}_{\text{n}}, b + 1, \#\} = a\{\underbrace{a, a, \cdots, a}_{\text{n}}, b, \#\}$$
 * $$a\{b+1, \#\} = a^a\{b, \#\}$$

Others
I found that: $$a\{0, b\}$$ represents $$a$$ inscribed in a $$b+3$$-gon in Steinhaus-Moser Notation. Therefore,
 * $$\text{Mega} = 2\{0, 2\} = 256\{256\}$$
 * $$\text{Moser} = 2\{0, 2\{0, 2\}-3\} = 2\{0, \text{Mega}-3\}$$

Bal Series
Unibal

$$\text{Unibal} = 1\{1\} = 1$$

Yes, I had to define this. :)

Bibal

$$\text{Bibal} = 2\{2, 2\}$$

This is equal to 256 inside a pentagon.

Tribal

$$\text{Tribal} = 3\{3, 3, 3\}$$

Yay, it's in the dictionary! LOL

Quadribal

$$\text{Quadribal} = 4\{4, 4, 4, 4\}$$

Skip a few ...

Decabal

$$\text{Decabal} = 10\{10, 10, 10, 10, 10, 10, 10, 10, 10, 10\}$$

King Bal

$$\text{King Bal} = 2218\{\underbrace{2218, 2218, \cdots, 2218}_{2218}\}$$

I like this one. XD

That's it for now. In the next part, I will redefine these numbers to make them look simpler. I'll post Part 2 sometime later. :)