User blog:TechKon/Some "function" i made

So basically, I came up with this function E(n) based mainly on my Mixed factorial. I tried my hardest to well-define this thing but I would appreciate any critiques and spotting of ill-definedness that maybe I can clear up. I will admit this is definitely all jumbled up and messy (both the layout of the rules and the function itself, haha) so I'm hoping I can clean this up a bit before I make it an official thing. I might just change E(2) to equal 2 so it can follow the rules of every other E(n) case besides E(1), so E(1) would the only 'base case'. Note that I call this "exploding mixed factorial function" which means I got the idea of this function from my Mixed factorial but this function isn't hardly a type of factorial. Anyways, let's get to it:

First, we define n and +n notation:

 n = any number in the infinite set ℕ n ≠ 0 n = any number in the infinite set ℤ+ (all positive integers) The notation of a plus with a superscript, +n is defined as the nth hyperoperator, starting from addition as n=1; +1 = + For instance, where n=3, x +3 y = xy  This function can be defined as follows:  For any case E(n), a sequence always begins with E(1) = 1. E(0) = undefined; n ≠ 0</li>  E(1) = 1 </li> E(2) = 3</li> These particular values I like to call 'base cases'. I will get to them in a few bullet points.</li></ul></li>  Always left-parenthesized: (((..)..)..) </li>  E(n) = ((( . . (((E(1) +<sup style="background-color:transparent">(E(1)) 2) +<sup style="background-color:transparent">(E(2)) E(2)) +<sup style="background-color:transparent">(E(3)) E(3)) +<sup style="background-color:transparent">(E(4)) E(4)). . . . )) +<sup style="background-color:transparent">(E(n-1)) E(n-1)) =  '''(((. . ((1 +<sup style="background-color:transparent">1 2) +<sup style="background-color:transparent">3 3) +<sup style="background-color:transparent">27 27) +<sup style="background-color:transparent">(E(4)) E(4))'''. . . . )) +<sup style="background-color:transparent">(E(n-1)) E(n-1))</li> When n &gt; 2, E(n) = E(n-1) +<sup style="background-color:transparent">(E(n-1)) E(n-1); or more simply: E(n) = E(n-1) +<sup style="background-color:transparent">(E(n-1))+1 2 E(1) ≠ E(1-1) +<sup style="background-color:transparent">(E(1-1)) E(1-1) = E(0) +<sup style="background-color:transparent">(E(0))  E(0) because E(0) has n = 0 which is undefined; this incorrect function would total out to 0, E(1) ≠ 0</li> E(2) ≠ E(2-1) +<sup style="background-color:transparent">(E(2-1))  E(2-1) = E(1) +<sup style="background-color:transparent">(E(1))  E(1) because E(1) is defined as 1; this incorrect function would total out to 2, E(2) ≠ 2</li>  E(1) and E(2) are 'base cases' since they build the beginning of any function sequence and are calculated by a different pattern rather than E(n) = E(n-1) +<sup style="background-color:transparent">(E(n-1)) E(n-1), or simply any other E(n) case. E(1) is the only case of this function that is defined as following the pattern where n=1: E(n) = n</li> E(2) is the only case of this function that is defined as following this pattern where n=2: E(n) = E(n-1) +<sup style="background-color:transparent">(E(n-1)) n =  E(1) +<sup style="background-color:transparent">(E(1)) 2 = 1+2 = 3</li></ul></li></ul></li></ul>Some cases of E(n) as examples: <li>E(0) = undefined; n ≠ 0</li> <li> E(1) = 1 </li> <li>E(2) = 3 = E(1) +<sup style="background-color:transparent">(E(1)) 2 = 1 + 2</li> <li>E(3) = 27 = (E(1) +<sup style="background-color:transparent">(E(1)) 2) +<sup style="background-color:transparent">(E(2)) E(2) = (1+2) +<sup style="background-color:transparent">(E(2)) E(2) = (1+2) +<sup style="background-color:transparent">(1+2) (1+2) = 3 +<sup style="background-color:transparent">3 3 = 33 = 27</li> <li>E(4) = ((E(1) +<sup style="background-color:transparent">(E(1)) 2) +<sup style="background-color:transparent">(E(2)) E(2)) +<sup style="background-color:transparent">(E(3)) E(3) = E(3) +<sup style="background-color:transparent">(E(3)) E(3) = 27 +<sup style="background-color:transparent">27 27 = 27 <span style="color:rgb(84,84,84);font-family:Roboto,arial,sans-serif;font-size:small">↑25 27</li> <li>and so on..</li></ul>