FANDOM


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 \(\varepsilon_1\)

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