Goobolthrex

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The goobolthrex is equal to \(a_{a_1}\) in the following sequence:

Let \(a_1\) = {10, 100 (1) 2} = goobol

Let \(a_2\) = {10, {10, {10, … {10, 100 (1) 2} (1) 2} (1) 2} … (1) 2} (\(a_1\) times) = gooboldex

Let \(a_3\) = {10, {10, {10, … {10, 100 (1) 2} (1) 2} (1) 2} … (1) 2} (\(a_2\) times) = gooboldudex

Continue, letting \(a_{n+1}\) = {10, {10, {10, … {10, 100 (1) 2} (1) 2} (1) 2} … (1) 2} (\(a_n\) times)

Goobolthrex = \(a_{a_1}\)

The term was coined by HaydenTheGoogologist2009.

Approximations

Notation	Approximation
Bowers' Exploding Array Function	\(\lbrace10,\text{goobol}+1,3(1)2\rbrace\)
Bird's array notation	\(\lbrace10,\text{goobol}+1,3[2]2\rbrace\)
DeepLineMadom's Array Notation	10[3{2}2]goobol + 1
Cascading-E notation	\(E100\#\text{^}\#100\#100\#(2+E100\#\text{^}\#99)\)
Strong array notation	s(10, goobol + 1, 3 {2} 2)
Fast-growing hierarchy	\(f_{\omega^{\omega}+2}^{f_{\omega^{\omega}}(100)}(100)\)
Hardy hierarchy	\(H_{\omega^{\omega^{\omega}}+2}^{H_{\omega^{\omega^{\omega}}}(100)}(100)\)