Multiexpansion

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.Array Notation by Jonathan Bowers
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 \(\{a,3,2,2\} = a\{a...a\{a\}a...a\}a (A\ a's)\)
*\(\{3,2,2,2\} = 3\{\{2\}\}2 = 3\{\{1\}\}3 = \{3,3,1,2\}\)
*\(\{4,2,2,2\} = \{4,4,1,2\}\)
*\(\{3,3,2,2\} = 3\{\{2\}\}3 = 3\{\{1\}\}3\{\{1\}\}3 = \{3,\{3,3,1,2\},1,2\}\)
*\(\{6,4,2,2\} = 6\{\{2\}\}4 = 6\{\{1\}\}6\{\{1\}\}6\{\{1\}\}6 = \{6,\{6,\{6,6,1,2\},1,2\},1,2\}\)
*\(\{10,100,2,2\} = \{10,\{10,99,2,2\},1,2\}\) (Kil-Doogol, Cookiefonster called this number mulporal)
== 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
== Sources ==
== See also ==
ja:乗算膨張
zh:Multiexpansion
Category:Functions
Category:Binary operators
Category:BEAF
Category:Jonathan Bowers
Category:Extended hyper operators