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Let a...[b] = aa...aa/w b a's. Then a....[b] = a...[a...[...[a...[a...[a]]]...]]/w b ...'s, a....[b] = a....[a....[...[a....[a....[a]]]...]]/w b ....'s, etc.

Let change a....[b] = a...[b,2], a.....[b] = a...[b,3], etc.

So a...[b,c,d+1] = a...[b,a...[b,...[a...[b,a...[b,a,d],d]]...,d],d] /w b ...'s, a...[b,c,d,e+1] = a...[b,c,a...[b,c,...[a...[b,c,a...[b,c,a,d],d]]...,d],d] /w b ...'s, etc.

Support a...[b[2]c] = a...[b,b...b,b]/w c b's, a...[b[2]c[2]d] = a...[b[2]c,c...c,c]/w d c's, a...[b[3]c] = a...[b[2]b...b[2]b]/w c b's, etc.

Finally, we have a...[b[0,1]c] = a...[b[c]b], a...[b[1,1]c] = a...[b[0,1]b...b[0,1]b]/w c b's, a...[b[0,2]c] = a...[b[c,1]b], a...[b[0,0,1]c] = a...[b[c,c]b], etc.

The limit is a...[a[a[...]a]a]/w a nested = a...[a[1][1]a]

So a...[x[1][1]y] = a...[x[x[...[x[x]x]...]x]x]/w y nested, a...[x[1][1]y[1][1]z] = a...[x[1][1]y[x[1][1]y[...[x[1][1]y[x[1][1]y]y]...]y]y]/w z nested, etc.

Then a...[x[1][2]y] = a...[x[x[...[x[x][1]x][1]...][1]x][1]x]/w y nested, a...[x[1][1][1]y] = a...[x[x][x[x][...[x[x][x]x]...]x]x]/w y nested, etc.

The limit is a...[a[a]_[a]...[a]_[a]a]/w a nested = a...[a[1][1]a].

Then a...[x[1][1]y] = a...[x[x]_[x]...[x]_[x]x]/w y [x]'s, a...[x[1]([1][1])y] = a...[x[1][x[1][...x[1][x]x...]x]x]/w y [1]'s

AndSoOn(x) = x...[x[1]([1][1]([1]...[1]([1][1][1])...))x]/w x*2 [1]'s

GoogolASN = AndSoOn(Googol)

Rules

Let # is rest of array and #_2 is seperators or comma.

  1. a...[b] = aa...aa/w b a's
  2. a...[# b,c] = a...[# a...[# ...[a...[# a...[# a,c-1],c-1]]...,c-1],c-1]/w b ...'s
  3. a...[# b[x+1 #]c] = a...[# b[x #]b...b[x #]b]/w c b's
  4. a...[# b[0 #_2 0...0,x+1 #]c] = a...[# b[c #_2 c...c,x #]b]
  5. More rules coming soon.

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