User blog comment:Syst3ms/Breadth-Recursive Array Notation (BRAN)/@comment-25601061-20180424215049/@comment-30754445-20180425162050

It's not random at all.

In the original notation, the 3rd entry adds an exponential. The 4th adds a double-arrow. The 5th adds a triple-arrow and so on. So yes, the last entry is the one that contributes most to the size of the number:

The first two entries don't do much at all:

B(2,2,2,2) ≈ 10↑↑1010 12.

B(3,2,2,2) ≈ 10↑↑1010 49

B(4,2,2,2) ≈ 10↑↑1010 134

B(2,3,2,2) ≈ 10↑↑1010 183.

B(2,4,2,2) ≈ 10↑↑1010 5863

The final exponent changes, but other than that nothing happens. All these numbers are between 10↑↑10↑↑3 and 10↑↑10↑↑4.

Even if go to B(1000000,1000000,2,2) it would still be below 10↑↑10↑↑5. Seems we have reached an impass, right?

But note how things change when we increment the 3rd entry:

(2,2,2,2) ≈ 10↑↑1010 12 . ≈ 10↑↑10↑↑3

(2,2,3,2) ≈ 10↑↑1010 10 12   ≈ 10↑↑10↑↑4

(2,2,4,2) ≈ 10↑↑1010 10 10 12  ≈ 10↑↑10↑↑5

See? (2,2,4,2) is already bigger than B(1000000,1000000,2,2)

And if we had (2,2,1000000,2) then:

(2,2,1000000,2) ≈ 10↑↑10↑↑1000001

Which is far bigger.

Now watch what happens when we increment the last entry:

(2,2,2,2) ≈ 10↑↑10↑↑3

(2,2,2,3) ≈ 10↑↑10↑↑10↑↑3

Whoa! See what just happened? This last number is comparable to (2,2,10↑↑10↑↑3,2). That's how much stronger the 4th entry is.

Continuing, we have:

(2,2,2,4) ≈ 10↑↑10↑↑10↑↑10↑↑3 ≈ 10↑↑↑5

(2,2,2,5) ≈ 10↑↑10↑↑10↑↑10↑↑10↑↑3 ≈ 10↑↑↑6

(2,2,2,6) ≈ 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑3 ≈ 10↑↑↑7

(2,2,2,7) ≈ 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑3 ≈ 10↑↑↑8

Now that's power!

And again, it doesn't matter at all what the other numbers are. In general:

(x,y,z,n) ≈ 10↑↑↑(n+1)

Similarly, we can show that:

(x,y,z,t,n) ≈ 10↑↑↑↑(n+1)

(x,y,z,t,u,n) ≈ 10↑↑↑↑↑(n+1)

and so on.

The exact same pattern happens with your new version. If you're starting with:

B(a,b,c) = a [c arrows] b

Then the 3rd entry counts the arrows (by definition)

And the 4th entry counts the layers of arrows.

And the 5th entry counts the groups of layers of arrows, and so on.

So again, the last entry contributes the most.

B(2,2,2,64), by the way, would be pretty close to Graham's Number in this version.This is because the Graham's function is 1 level stronger than arrows, and your rules result in going up 1 level for every additional entry.