User blog comment:Syst3ms/Breadth-Recursive Array Notation (BRAN)/@comment-25601061-20180424215049/@comment-30754445-20180425121217

Notations are analyzed by carefully calculating the actual value of several of arrays, starting from the simplest ones and going up.

You'll need some kind of standard notation for large numbers for this to work. Knuth arrows will do nicely for this notation. Or if you're familiar with the FGH then you can use that as well.

At any rate, 3-arrays seem to have tetrational growth:

B(x,y,z) ≈ x↑↑z

B(2,2,2,2) is about 10↑↑1010 12.

And is looks like the last entry is the only one that makes the numbers significantly larger. It seems that:

B(x,y,z,2) remains stuck at about 10↑↑1010 12 .for reasonable values of x,y,z.

B(x,y,z,3) then jumps to 10↑↑10↑↑1010 12.

And in general B(x,y,z,t) ≈ 10↑↑↑t, so four arrays give us pentational growth.

So B(2,2,2,2,2) would be a large pentational number:

B(2,2,2,2,2) ≈ 10↑↑↑B(2,2,2,3) ≈ 10↑↑↑10↑↑10↑↑1010 12

B(2,2,2,2,2,2) ≈ 10↑↑↑↑B(2,2,2,2,3) ≈ 10↑↑↑↑10↑↑↑10↑↑↑10↑↑10↑↑1010 12

B(2,2,2,2,2,2,2) ≈ 10↑↑↑↑↑B(2,2,2,2,2,3) ≈ 10↑↑↑↑↑10↑↑↑↑10↑↑↑↑10↑↑↑10↑↑10↑↑1010 12

and so on.

In general, an n-entry array whose final entry is k will be comparable to:

10[n-1 arrows]k

So the limit of BRAN is the same as the limit of knuth up-arrows. It is an ω-level function (which means that it isn't powerful enough to represent Graham's Number)