Finalsonic Triangle Notation

The final extension to the Ultrasonic Triangle Notation.

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

Tr@@@(n) =Tr@@n(n)

Tr@@@0(n) -> Again, just like before add 1 to "n" in the subscript like you did with 2 @@'s

The same iteration rule for @@@1, just like with @@ but of course now we have 3.

The same for however many @'s

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

In the last notation, we had a rule identical to this one. However, we are going to extend it here. By a lot.

You see, the next rule would be to define &1 and multiple &'s. And they would have the same iteration rules as the ones above, except replace everything with the last step. (I think you know what I mean).

We first had #, then @ and now &.

Let's define !n.

!n would be defined as the nth triangularion.

!1 = #

!2 = @

!3 = &

!4 = The next step which would be an iteration of &'s just like the way @ iterated #'s.

Tr!4(n) = Tr&&&&&....&&&(n), with "n" &'s.

Now,

Tr!!(n) = Tr!Tr ! n(n) (Tr!Tr ! n(n)  (...Tr!Tr ! n(n) (n))..)), iterated Tr!Tr ! n(n)  (n) times.

Tr!!!(n) = Tr!!(Tr!!(...Tr!!(n))..)), with Tr!!(n) iterations.

!->m = !!!!....!!! (m iterations)

Tr!->m(n) = Tr!!!...!!!! (m !'s)(n) = Tr!!!...!!!! (m-1 !'s)(Tr!!!...!!!! (m-1 !'s)(...Tr!!!...!!!! (m-1 !'s)(n))..)), iterated Tr!!!...!!!! (m-1!'s)(n) times

Then,

Tr?(n) = Tr!->n+1(n)

Tr??(n) = Tr?(Tr?(...Tr?(n))..))

?->m = ???...??? (m iterations)

Tr?->m = Same iteration as with !->m, but replace the !'s with ?'s.

Tr¿(n) = Tr?->n+1(n)

Same iteration rules as with the ?.

And we're almost done.

Define "^n" as the nth iterator.

Eg: ^1 = !

^2 = ?

^3 = ¿

^4 = An iteration of "¿" just like with "?"

Finally, define Trω(n) as Tr^n+1(n)

Additional rule Edit
Additional rule that is not part of the original notation. If it is well defined, then it is. For now, it remains experimental until it is confirmed to be well defined.

You see how we have been iterating things since "!"?

Define TrΩ(n) as the Trω(n)th step of iteration for the entire notation.