User blog:Nayuta Ito/Multidimentional Ackermann Function (Part1)

These are the definitions:

X:0 or more integers and 0 or more (n)'s Y:0 or more zeroes and 0 or more (n)'s Z:1 or more zeroes and 0 or more (n)'s  Commas can be replaced by (1). 1.{Ya}=a+1 2.{Xb+1,0}={Xb,1} 3.{Xb+1,a+1}={Xb,{Xb+1,a}} 4.{Xb+1,0Ya}={Xb,aYa} 5.{Xb+1(n+1)a}={Xb(n+1)a(n)…a times a's…(n)a} 6.{Xb+1(n+1)ZYa}={Xb(n+1)a(n)…a times a's…(n)aYa} (Z does not contain (n+1))

This notation reaches up to $$f_{\omega^{\omega^{\omega}}}(x)$$ and this is posted:

多次元アッカーマン配列

Before I want to extend it, I found that it has very interesting property.

This looks like an "base-omega" number like this:

$$\{ a_n, a_{n-1}, \cdots a_1, a_0, x \} \simeq f_{a_n \omega^n + a_{n-1} \omega^{n-1} +\cdots + a_1 \omega^1 + a_0}(x) $$

This relation continues up to n dimension. For example, the rightmost number in the second row is $$\omega^{\omega}$$