Basic array notation: hardly even an array notation! Just the expression a[c]b, which is equal to the hyperoperator starting from addition, multiplication, exponentiation, and so on. FGH level ω.
The basic array notation has the following form: a[c]b. Note that the function is only defined for any natural numbers a and b greater than 0, and c can be either positive integers or zero. for c = 0, this is only defined for the basic array notation.
To solve the expression, we need some rules as follows:
Successor Rule: a[0]b = a+1
Base Rule: a[1]b = a+b, a[2]b = ab, a[3]b = a^b
Prime Rule: a[c]1 = a
Recursion Rule: a[c]b = a[c-1](a[c](b-1)), provided c ≥ 0 and b ≥ 1.
It's easy to see that the above three rules mirror the definition of up-arrow notation. So the limit of this notation is f_ω(n).
A small example:
2[5]3 = 2[4]2[4]2 = 2[4]2[3]2 = 2[4]2^2 = 2[4]4 = 2[3]2[3]2[3]2 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536.
[1] function stands for addition (a+b),
[2] function stands for multiplication (ab),
[3] function stands for exponentiation (a^b),
[4] function stands for tetration (4-ation, a^^b),
[5] function stands for pentation (5-ation, a^^^b),
[6] function stands for hexation (6-ation, a^^^^b),
And so on.