User blog:MachineGunSuper/Hypersonic Triangle Notation

So, recently I noticed that a while ago I made a function that I wanted to post on this site but didn't have the motivation to. I have decided coming back to it, so here it is. Hypersonic Triangle Notation (HTN)

Define a//b as '''ab a b a. . . a b       ''', where the combination ab repeats itself many times, basically a//b = ab↑↑ab.

Now, define t(n) as "n↑↑(n//nn)//n↑↑(n//nn-1)//...//n↑↑(n//nn-n)  //  (n//n+1)//(n//n+2)//(n//n+3)//...//(n//nn)"

Eg: t(3) =(3↑↑(3//27)//3↑↑(3//26)//3↑↑(3///24))//(3//4)//(3//5)//....(3//27)

Now,

△(n) = t(t(t(...(t(n))))...)), where there are t(t(t(...(t(t(n))) t's (1 less t), and keep going until you reach t(t(t(t....(t(n)))) where there are t(n) t's

△△(n) = △(△(△(....(△(n))))), and the same thing here where in each expression there is one less △ until you reach △(n) triangles.

△△△(n) = iteration of △△(n) the same way △(n) was iterated before

and so on

Then, △m (n) = △△.....△△ (n) with m △'s

△m,p (n) = △m (△m (△m (....△m (n)))...))), repeated P times

△m,p,a (n) = △m,p (△m,p (△m,p (....△m,p (n))))...))), repeated a times

etc..

b△a(n) = △a,a,a,a,a.....,a,a,a(n), where "a" is repeated b times

b,c△a (n) = b△a(b△a(b△a(....b△a(n))))...))), iterated "c" times

b,c,d△a(n) = b,c△a (b,c△a (....b,c△a (b,c△a (n))..)), iterated "d" times

etc


 * Also, having more elements on both sides is currently yet to be defined, so for example there is no b,c△a,m(n) YET. (you can only have more terms on one side)

b△ca(n) = b,b,b,b...,b,b,b△a(n), where "b" is iterated c times.

A golden rule is that, if you want to have the element at the top of the triangle, you must also have the ones at the bottom (pretty obvious)

b△c,ma(n) = b△ca(b△ca(b△ca(....b△ca(n)))...)), iterated "m" times.

Etc..

Finally,

Define T△(n) as b△ca(n) △b△ ca(n)    b△ca(n)  (n)

LaTeX may have some trouble picking that up, so what T△(n) means is having  " n△nn(n) " on all 3 sides of the triangle.