User:A()/How Big Can We Go?

User:A() created a monster, a monster of brackets.

Syntax

A+1=(A)A
1 = ()

Expansion

First, we have two brackets:

()

Then we expand:

(())()

And we can keep expanding, forever. Until your computer crashes. We had one pair, then we had a pair in a pair, connected to a pair. Three pairs. Next:

((())())(())()

4 pairs plus 3 pairs; 7 pairs. Let's expand again:

(((())())(())())((())())(())()

8 pairs plus 7 pairs; 15 pairs. The pattern looks like this:

${\displaystyle N\left(a\right)=N\left(a-1\right)+N\left(a-1\right)+1}$

$N\left(1\right)=1$

or

$N\left(a\right)=2N\left(a-1\right)+1$

$N\left(1\right)=1$

How Big Can We Really Go?

Here is a big group of pairs:

User:A()/How Big Can We Go?/Large Group of Pairs

Yes, it's humongous, but we can go bigger, how bigger? Well, the average Window's text file can store up to 16 Tib - 64 KiB, or 1.0995E+12 Chars per text file! Or I guess per computer since Windows has up to about 1 TiB of storage...