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Multiexpansion refers to the binary function $$a \{\{2\}\} b = \{a,b,2,2\} = \underbrace{a \{\{1\}\} a \{\{1\}\} \ldots \{\{1\}\} a \{\{1\}\} a}_{\text{b a's}}$$ using BEAF.[1]

In the fast-growing hierarchy, $$f_{\omega+2}(n)$$ is comparable to multiexpandal growth rate, which means multiexpansion is comparable to $$a\rightarrow a\rightarrow b\rightarrow 3$$ in Chained Arrow Notation and $$(a\{3,3\}b)$$ in Notation Array Notation.

## Examples

• $$\{a,3,2,2\} = a\{\{1\}\}a\{\{1\}\}a$$. This is equal to a expanded to (a expanded to a). Let $$A = a\{a\{a...a\{a\{a\}a\}a...a\}a\}a (a\ a's)$$, then $$\{a,3,2,2\} = a\{a...a\{a\}a...a\}a (A\ a's)$$
• $$\{3,2,2,2\} = 3\{\{2\}\}2 = 3\{\{1\}\}3 = \{3,3,1,2\}$$
• $$\{4,2,2,2\} = \{4,4,1,2\}$$
• $$\{3,3,2,2\} = 3\{\{2\}\}3 = 3\{\{1\}\}3\{\{1\}\}3 = \{3,\{3,3,1,2\},1,2\}$$
• $$\{6,4,2,2\} = 6\{\{2\}\}4 = 6\{\{1\}\}6\{\{1\}\}6\{\{1\}\}6 = \{6,\{6,\{6,6,1,2\},1,2\},1,2\}$$

## Pseudocode

Below is an example of pseudocode for multiexpansion.

function multiexpansion(a, b):
result := a
repeat b - 1 times:
result := expansion(a, result)
return result

function expansion(a, b):
result := a
repeat b - 1 times:
result := hyper(a, a, result + 2)
return result

function hyper(a, b, n):
if n = 1:
return a + b
result := a
repeat b - 1 times:
result := hyper(a, result, n - 1)
return result