f notation

$f_0(a,b)=a^b$

and

$f_0(a,b,c)=a^{b^c}$

so

$f_0(a[0]2)=f_0(a,a)=a^a$

And$f_0(a[0]10)=a\uparrow \uparrow 10$

Also$f_0(a[0]b,c)=f_0(a,a...a,a,c)$ with b copies of a

$f_0(a[1]2)=f_0(a[0]2) \uparrow\uparrow2 = f_0(f_0(a,a),f_0(a,a))$

$f_0(a[2]3)=((f_0(a,a,a))\uparrow\uparrow3)\uparrow\uparrow3$

$f_0(a[c]b)=(f_0(a[c-1]b))\uparrow\uparrow b$

This is going to get cumbersome soon, so i define $f_1$

$f_1(a,b,c)=f_0(a[c]b)$

$f_1(a,b,c,d)=f_1(a,b,f_1(a,b,c,d-1))$

Primitive recursive!

$f_1(a,b,c,d,e)=f_1(a,b,c,f_1(a,b,c,d,e-1))$

And it goes on... forever

So:

$f_1(a[0]b)=f_1(a,a...a,a)$with b repititions.

$f_1(a[1]b)=f_1(f_1(a,a...a,a),...,f_1(a,a...a,a))$with b repititions of 'a' for the first f_1,then,put the f_1 block and there are b repititions of that block.

In general.

$f_x(a[b]c)=f_x(f_x(a[b-1]c),...,f_x(a[b-1]c))$with c repititions and with ALL whole number X that is not negative

$f_x(a[0]b)=f_x(a,a...a,a)$b repeats.

and

$f_{x-1}(a[c]b)=f_x(a,b,c)$

So the final form of my notation:

Weak F function

$f(a,b,c,d)=f_d(a,b,c)$

Strong F function

$F(a,b,c,d)=f_d(a[c]b)$