User blog:Googolplix/Linear Iterating array notation

The Linear Iterating Array Notation (lIAN) is the first part of my Iterating Array Notation (IAN)

Rules of lIAN:

2 entry IAN (1 rule):

- I(a,b,) = a^b

3 entry IAN (1 rule):

- I(a,b,c,) = I^b(a,a,c-1,)

Iteration IAN (2 rules):

- I^n+1(a,b,#) = I(a,I^n(a,b,#),#)

- I^1(#) = I(#)

lIAN (x rules):

- I(#,1,) = I(#)

- I(a,b,#,1,c+1,#) = I(a,a,#,b,c,#)

- I(a,1,#) = a

Examples:

I(3,2,2,) = I^2(3,3,1,) = I(3,I(3,3,)) = I(3,27,) = 7625597484987 > {3,2,2,}

I(3,3,2,2,) = I^3(3,3,1,2,) = I(3,I(3,I(3,3,1,2,),1,2,),1,2,) = I(3,I(3,I(3,3,3,),1,2,),1,2,) = I(3,I(3,I^3(3,3,2,),1,2,),1,2,)

I(3,I(3,I(3,I(3,I(3,3,2,),2,),2,),1,2,),1,2,) = I(3,I(3,I(3,I(3,I^3(3,3,),2,),2,),1,2,),1,2,) = I(3,I(3,I(3,I(3,I(3,3^^2,),2,),2,),1,2,),1,2,) = I(3,I(3,I(3,I(3,3^^3,2,),2,),1,2,),1,2,) = I(3,I(3,(3,I^3^^3(3,3,),2,),1,2,),1,2,) = I(3,I(3,I(3,3^^^3,2,),1,2,),1,2,) = I(3,I(3,3^^^4,1,2,),1,2,) = I(3,I(3,3,3^^^4,),1,2,) = I(3,3{3^^^4}3,1,2,) = I(3,3,3{3^^^4}3) = 3{3{3{3}4}3}3 > {3,3,2,2,}

I(3,3,1,1,2,) = I(3,3,1,3,) = I(3,3,3,2,) = I^3(3,3,2,2,) = I(3,I(3,3{3{3{3}4}3}3,2,2,),2,2,) = I(3,I^3{3{3{3}4}3}3(3,3,2,2,),2,2,) >> I(3,{3,3,1,3,},2,2,) = I^{3,3,1,3,}(3,3,2,2,) > I(3,{3,3,1,4,},1,2,) = I(3,3,{3,3,1,4,}) = 3{3{}3}3

Growth rate:

I(a,b...y,z,) (n entries) ≈ f⍵^⍵(n)

lIAN > Linear BEAF.