## FANDOM

10,825 Pages

Let change the rule: $$n![\circ 1,m+1 \bullet]$$ = $$n![\circ ...[\circ 1,m \bullet]...,m \bullet]$$ (n nested)

$$\circ$$ is rows of 1

$$\bullet$$ is anything

### Growth Rates

[[1]] has growth rate $$\omega 2$$

[[n]] has growth rate $$\omega 2 + (n-1)$$

[[[1]]] has growth rate $$\omega 3$$

[1,2] has growth rate $$\omega^2$$

[[1],2] has growth rate $$\omega^2 + \omega$$

[[1,2],2] has growth rate $$\omega^2 2$$

[1,3] has growth rate $$\omega^3$$

[1,n] has growth rate $$\omega^n$$

[1,[1]] has growth rate $$\omega^\omega$$

[1,[1,2]] has growth rate $$\omega^{\omega^2}$$

[1,1,2] has growth rate $$\varepsilon_0$$

[[1,1,2],1,2] has growth rate $$\varepsilon_0 2$$

[1,2,2] has growth rate $$\varepsilon_0 \omega$$

[1,[1],2] has growth rate $$\varepsilon_0 \omega^\omega$$

[1,[1,1,2],2] has growth rate $$\varepsilon_0^2$$

[1,[1,[1],2],2] has growth rate $$\varepsilon_0^2 \omega^\omega$$

[1,[1,[1,1,2],2],2] has growth rate $$\varepsilon_0^3$$

[1,1,3] has growth rate $$\varepsilon_0^\omega$$

More growth rates soon!

## Even Stronger

[1,2] has growth rate $$\omega^2$$

[[[1,2]],2] has growth rate $$\omega^2 2$$

[[1,2],2] has growth rate $$\omega^3$$

[[[[1,2]],2],2] has growth rate $$\omega^3 + \omega^2$$

[[[[[1,2],2]],2],2] has growth rate $$\omega^3 2$$

[[[1,2],2],2] has growth rate $$\omega^4$$

[1,3] has growth rate $$\omega^\omega$$

[[1,2],3] has growth rate $$\omega^{\omega + 1}$$

[[[1,3],2],3] has growth rate $$\omega^{\omega 2}$$

[[[[[1,3],2],3],2],3] has growth rate $$\omega^{\omega 3}$$

[[1,3],3] has growth rate $$\omega^{\omega^2}$$

[[[1,3],3],3] has growth rate $$\omega^{\omega^3}$$

[1,4] has growth rate $$\omega^{\omega^\omega}$$

[[1,2],4] has growth rate $$\omega^{\omega^\omega + 1}$$

[[[1,3],2],4] has growth rate $$\omega^{\omega^\omega + \omega}$$

[[[1,4],2],4] has growth rate $$\omega^{\omega^\omega 2}$$

[[1,3],4] has growth rate $$\omega^{\omega^{\omega + 1}}$$

[[[1,4],3],4] has growth rate $$\omega^{\omega^{\omega 2}}$$

[[1,4],4] has growth rate $$\omega^{\omega^{\omega^2}}$$

[1,5] has growth rate $$\omega^{\omega^{\omega^\omega}}$$

[1,[1]] has growth rate $$\varepsilon_0$$

[[1],[1]] has growth rate $$\varepsilon_0 + \omega$$

[[[1,[1]]],[1]] has growth rate $$\varepsilon_0 2$$

[[1,2],[1]] has growth rate $$\varepsilon_0 \omega$$

[[[[[1,2],[1]]],2],[1]] has growth rate $$\varepsilon_0 (\omega 2)$$

[[[1,2],2],[1]] has growth rate $$\varepsilon_0 \omega^2$$

[[[1,3],2],[1]] has growth rate $$\varepsilon_0 \omega^\omega$$

[[[1,[1]],2],[1]] has growth rate $$\varepsilon_0^2$$

[[1,3],[1]] has growth rate $$\varepsilon_0^\omega$$

[[[1,4],3],[1]] has growth rate $$\varepsilon_0^{\omega^\omega}$$

[[1,4],[1]] has growth rate $$\varepsilon_1$$