User blog comment:Primussupremus/Extending my notation past the 2nd level and into the 3rd./@comment-28606698-20170420181215

Let me help you, Prim. We should define rule set, to avoid confusing. For example as follows:

1) a,b,c are integer variables and if b less than or equal to 56 then it can be marked by letter of english alphabet a..Z

2) a,0,c=a+b

3) a,b,0=1 for b>2

4) a,b,c=a,b-1,(a,b,c-1)

5) a,b,c#d=a,(a,b,c#d),c

6) a,b,c#(d)=a,b,c#a,b,c#(d-1)

7) a,b,c#[d]=a,b,c#(a,b,c#[d-1])

I think this rule set is what you meaned in your posts. If apply this rules then

a,a,a is approximately equal to f_w(a)

a,a,a#a is approximately equal to f_w+1(a)

a,a,a#(a) is approximately equal to f_w+2(a)

a,a,a#[a] is approximately equal to f_w+3(a)

Is not it?