Expansion

Expansion refers to the function \(a \{\{1\}\} b = a \{a \{\cdots \{a\} \cdots\}a\}a\), where there are b a 's from the center out. It is {a,b,1,2} in BEAF. The notation a{c}b means {a,b,c}, which is a "c + 2"-ated to c.

It grows faster than any hyper-operator, such as tetration or pentation. In the fast-growing hierarchy, \(f_{\omega+1}(n)\) corresponds to expandal growth rate.

Graham's number is defined using a very close variant of expansion. It is \(3 \{\{1\}\} 65\) with the central 3 replaced with a 4.

Examples
\(2\ \{\{1\}\}\ 2 = \{2,2,3\} = 2 \uparrow\uparrow\uparrow 2 = 4\)

\(3\ \{\{1\}\}\ 2 = \{ 3,2,1,2 \} = 3 \{3\} 3 = 3\uparrow\uparrow\uparrow 3\) (tritri)

\(3\ \{\{1\}\}\ 3 = \{3,3,1,2\} = 3 \{3 \{3\} 3\} 3 = 3\{\text{tritri}\}3 = 3\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{tritri}}3\)