X-Sequence Hyper-Exponential Notation is an extension of the hyper operators by SuperJedi224[1], partly inspired by both Cascading-E notation and BEAF.
Definition[]
Main Definition
m{1}n = mn
m{α}1 = m
m{α+1}(n+1) = m{α}(m{α+1}n)
If α is a limit case: m{α}n = m{α[n]}m
Definition of α[n]
X[n] = n
(α+β)[n] = α+β[n]
(α*(β+1))[n] = α*β+α[n]
(α*β)[n] = α*β[n], if β is a limit case
α*1 = α
(αβ+1)[n] = αβ*α[n]
(αβ)[n] = αβ[n], if β is a limit case
α1 = α
(α↑↑X)[0] = 1
(α↑↑X)[n] = α(α↑↑X)[n-1]
α>(β+1) = (α>β)↑↑X
(α>β)[n] = α>β[n], if β is a limit case.
α>0 = α
(α>>X)[0]=0
(α>>X)[n+1]=α>(α>>X)[n]
Growth Rate[]
This notation is believed to have a limit ordinal of \(\zeta_1\) in the Fast-growing hierarchy.
Examples[]
4{3}7 = 4{2}4{2}4{2}4{2}4{2}4{2}4 (this is solved from right to left).
4{X+1}3 = 4{X}4{X}4 = 4{X}4{4}4 = 4{4{4}4}4.
4{X>X>X}3 = 4{X>X>3}4 = 4{X>((X↑↑X)↑↑X)↑↑X)}4
In Other Notations[]
a{c}b = \(a\uparrow^cb\)
a{X}b = \(a\uparrow^ba\)
a{X+1}b = {a,b,1,2}
a{X+2}b = {a,b,2,2}
a{X*2}b = {a,a,b,2}
a{X2}b = {a,a,a,b,2}
a{X↑↑2}b=a{XX}b ~ {a,b(1)2}
a{X↑↑3}b=a{XXX}b ~ {a,b(0,1)2}
a{X↑↑4}b=a{XXXX}b ~ {a,b((1)1)2}
a{X↑↑X}b ~ {a,b[1\2]2}
a{(X↑↑X)X}b ~ {a,b[2\2]2}
a{(X↑↑X)(X↑↑X)}b ~ {a,b[1[1\2]2\2]2}
a{(X↑↑X)↑↑X)}b ~ {a,b[1\3]2}
a{X>X}b ~ {a,b[1\1,2]2}
a{X>X>X}b ~ {a,b[1\1[1\2]2]2}
a{X>>X}b ~ {a,b[1\1\2]2}