User blog:GamesFan2000/My new notation

The following notation is called the Expanding Separator Notation. It takes the basic form of n[X]m.

Terminology
The base, n, is the variable that is “acted upon” in the operation.

The expander, m, determines the amount of expansions needed for a separator.

The separator, [X], is the level-based operator in this notation.

First-Level Separators
All expressions in this notation are solved left-to-right.

1 – n[0]0=n^nn

2 – n[0]1=n^nn[0]0[0]0[0]0...n^nn [0]’s...[0]0

3 – n[0]m+1=(n[0]m)[0]m[0]m[0]m...n[0]m [0]’s...[0]m

Examples
2[0]0=4, 2[0]1=4[0]0[0]0[0]0[0]0

3[0]3=(3[0]2)[0]2[0]2[0]2...3[0]2 [0]2’s...[0]2

Second-Level Separators
4 – n[1]0=(n[0]n)[0](n[0]n)[0](n[0]n)...n[0]n [0]’s...[0](n[0]n)

5 – n[1]m+1=(n[1]m)[1]m[1]m...n[1]m [1]’s...[1]m

Examples
4[1]0=(4[0]4)[0](4[0]4)...4[0]4 [0]’s...[0](4[0]4)

4[1]4=(4[1]3)[1]3[1]3...4[1]3 [1]’s...[1]3

Higher-Level Separators
6 – n[X+1]0=(n[X]n)[X](n[X]n)...n[X]n [X]’s...[X](n[X]n)

7 – n[X]m+1=(n[X]m)[X]m[X]m...n[X]m [X]’s...[X]m

Examples
3[4]0=(3[3]3)[3](3[3]3)...3[3]3 [3]’s...[3](3[3]3)

7[5]8=(7[5]7)[5]7[5]7...7[5]7 [5]’s...[5]7