Ultrasonic Triangle Notation

The Ultrasonic Triangle Notation is an extension to the Hypersonic Triangle Notation.

T0△(n) = T(n)

T1△(n) = T(T(T(...(T(n))..)), repeated T(n) times.

Ta+1△(n) = Ta(Ta(...(Ta(n))..))

Ta,b△(n) = TT a△(b) △(n)

Ta,b,c△(n) = TT T a,b△(c) (n) (n)

Ta,b,c,....,p,x△(n) (with "m" elements in the subscript) =

TT T ... T a,b,c,.....,p△(x) △(n).... △(n) △(n) △(n), where T repeates "m" times, aka the bottom T function is applied to x (the last element), then the rest "m-1" T's are applied to "n".

Let's introduce "The triangularions"

T#0△(n) = Tn+1△(n)

T#1(n) = Tn,n,n,....,n,n△(n), where there are T#0△(n) "n's" in the subscript.

For notational purposes, a T function can also be written as Tr(n) as it is annoying to keep pasting the △ in

.Tr#2(n) = Tr#1(Tr#1(...Tr#1(n))..)), iterated Tr#1(n) times.

Tr#m(n) = Tr#m-1(Tr#m-1(...Tr#m-1(n))..)), iterated Tr#m-1(n) times.

Tr##(n) = Tr#n(n)

Tr##0(n) = Tr#n+1(n)

Tr##1(n) = Tr##0(Tr##0(Tr##0(...Tr##0(n))..)), iterated Tr##0(n) times.

Tr##m(n) =Tr##m-1(Tr##m-1(...Tr##m-1(n))..)), iterated Tr##m-1(n) times.

Tr###(n) = Tr##n(n)

Tr###0(n) = The same rule, just add 1 to "n" just like with 2 #'s

The same iteration rules for ###1, just like with 2#'s but of course it has 3 this time.

The same goes for however many #'s you have.

One last thing (for now.)

Tr@(n) = Tr###...###(n), with "n" #'s.