User blog:Syst3ms/Breadth-Recursive Array Notation (BRAN)

Breadth-Recursive Array Notation
Hi there ! Today I'm gonna present an array notation I devised !

I really got into googology recently, and the idea of defining my own notation crossed my mind, so I was like "why not". After trying many things and realizing either : I finally came up with a quite simple yet powerful notation (I don't know how powerful, I have low expectancies anyway).
 * They were naive extensions
 * They recursed infinitely
 * They weren't powerful (unless they were naive extensions of other powerful notations)

So, the name's Breadth-Recursive Array Notation or BRAN for short. Yes, it's yet another array notation (I was tempted to make it a recursive acronym, actually). One of the few things particular with it is that it's not an extension of arrow notation (and I don't know how it would affect the notation if it were, actually) and that its strength relies heavily on how it collapses 1s.

Okay, let me just define it already : In general, a 1 increments all of its neighbours and disappears. This collapsing rule is evaluated from right to left.
 * Rule #1 : 
 * Rule #2 : 
 * Rule #3 :
 * Rule #4 : 
 * Rule #5 (the most important one) : 

Here are some example expansions :

B(2,2,2) :

B(2,2,2,2) :

The expansions only get more and more tedious from now on, so I'll do you a favor and tell you the final value is

$$81^{27^{136}}$$ If this value seems utterly random, then know it's a contraction of the longer :

$${(81^{27^8})}^{({27^8})^{16}}$$

If you're still not convinced, then feel free to evaluate the expression yourself.

Some special properties :

 * No BRAN array is degenerate, aside from . What that means is that there is no other array that will collapse very fast to a small value. For example,  in BAN for any n.
 * A few equalities :
 * for all n > 1
 * for all m > 1 and n > 1

Okay, so this is all for me, I guess. Thanks for reading !