User blog:Wythagoras/Dollar function part 2

Linear Array Notation
Normal brackets have level 1

1. a$b \(\bullet\) = (a+b)$ \(\bullet\)

2. a\(\circ\)[0 \(\bullet\)]\(\circ\) = a$\(\circ\)a[ \(\bullet\)]\(\circ\)

3. a$\(\circ\)[b+1\(\bullet\)]\(\circ\) = a$\(\circ\)[b\(\bullet\)][b\(\bullet\)]...[b\(\bullet\)][b\(\bullet\)]\(\circ\) a [b\(\bullet\)]'s Where 0 and b are the less nested numbers

4. a$[@]_(b+1)\(\bullet\) = a$[@]_b = a$[[...[[@]_b ]...]_b ]_b\(\bullet\)  where there are a b-brackets and the b-bracket is the bracket with the lowest level.

here contains \(\diamond\) only brackets with level bigger than b and zeroes

5. a$%[\(\diamond\)]% = a$%\(\diamond\)\(\diamond\)...\(\diamond\)\(\diamond\)% where there are a \(\diamond\)'s  ( here is 'b' 1 )

6. a$[b\(\bullet\)]_(c+1) = a$[a$[...a$[a$[b-1\(\bullet\)]_(c+1)\(\bullet\)]_(c)\(\bullet\)...]_(c)\(\bullet\)]_(c) \

7. \(\bullet\)0 = \(\bullet\)

8. a$[0,b] = a$[0,b-1]_[0,b-1]...[0,b-1]_[0,b-1]

9. a$[0,0...0,0,b \(\bullet\)] = a$[0,0...0,[0,0...0,[...[0,0...0,0,0 \(\bullet\)]...],b-1 \(\bullet\)],b-1 \(\bullet\)]