Multiexpansion refers to the 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] Or a expanded to a for b times.
In the fast-growing hierarchy, \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate.
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}
- {4,3,2,2} = {4,{4,4,1,2},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
Sources
See also
Main article: hyper operator
Hyper-operators: successor · addition · multiplication · exponentiation · tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulationBowers' extensions: expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation
Saibian's extensions: hexonation · heptonation · octonation · ennonation · deconation
Tiaokhiao's extensions: megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...