User blog:GamesFan2000/GamesFan's Array Function

I've known about this wiki for quite a while, and as a person who loves math, the stuff featured here is very interesting. So, I decided to try my hand at making a googological function. It's a unique form of array notation that I have theory crafted over the past few weeks with different rules and symbols. The language used in it is quite simple, and some of the definitions might not be obvious unless you apply earlier rules or are good at understanding ambiguity. I also do not have a good understanding of the fast-growing hierarchy or related ordinal-based functions, so I won't be providing comparisons. The only comparison I've included within my explanation is the three-entry arrays under my rules compared to regular arrays. This is my array notation, which you can call GamesFan's array notation, but I've called it Explosively-Powerful Array Notation, or EPAN. I'd like to see someone solve the growth of this function.

Linear Arrays

Part 1: Single-entry and Two-entry arrays

(a) = a

(a, b) = a^b

(3, 2) = 9

Part 2: Three-entry arrays

(a, b, c) = ((a, b, c-1), (a, b, c-1), c-1) = (((a, b, c-1), (a, b, c-1), c-2), ((a, b, c-1), (a, b, c-1), c-2), c-2)

(2, 2, 2) = (((2, 2, 1), (2, 2, 1)), ((2, 2, 1), (2, 2, 1))) = (((4, 4), (4, 4)), ((4, 4), (4, 4))) =

((256, 256), (256, 256)) = 256^256^256^256 = 256^^4 = {256, 4, 2}

Part 3: Beyond three entries

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

A(n) = (n, n, n, n, n, n, n…, n), or an n-length array of n-valued entries

Planar Arrays

Part 1: Condensed Linear Arrays

(a(b)c) = An array of a’s with a length of c^b

(3(2)3) = (3, 3, 3, 3, 3, 3, 3, 3, 3)

(3, 3(2)3, 3) = ((3, 3), (3, 3) …, (3, 3)), or a 27^2-length array of (3, 3) arrays

(a(b)c(d)e) = An e^d-length array of c^b-length arrays of a-valued entries

Part 2: Dimensional Arrays

(a (b, c) d) = A (d, b, c)-length array of a-valued entries

(2 (2, 2) 2) = A 256^^4-length array of 2’s

(a (b, c) d (e, f) g) follows the same rules as (a(b)c(d)e)

Part 3: Super-dimensional Arrays

(a (b, c, d) e) = An ((e, b, c, d), (e, b, c, d), (e, b, c, d), (e, b, c, d))-length array of a-valued entries

Above two planar definers, the arrays that define the length become contained within larger arrays, with the arrays becoming more and more contained; in other words, an array of arrays of arrays of arrays, so on and so forth

AP(n) = (n, n, n, n…, n (n, n, n, n…, n) n, n, n, n…, n (n, n, n, n…, n) …, n), or an n-length series of n-length planar definitions applied to n-length arrays, all with entries of n-value

Cuboidal Arrays

(a[b]c) = A c^^b-length series of c^^b-length planar definitions applied to c^^b-length arrays, all with a-valued entries

(a [b, c] d) = A (((d, b, c), (d, b, c), (d, b, c)) …) or array of arrays of (d, b, c) arrays-length series of array of arrays of (d, b, c) arrays-length planar definitions applied to array of arrays of (d, b, c) arrays-length arrays, all with a-valued entries

ACu(n) applies the same logic to cuboidal arrays as AP(n) does to planar arrays

The arrays for defining length are contained twice more than in planar for cuboidal

Should you go on to condensed cuboidal arrays, add an extra set of squared brackets around the definers, use pentation for single definers, and contain the definer arrays twice more than in cuboidal, and apply this same logic for all future condensations, adding one set of squared brackets, upping the strength of the operator by one for single-definers and increasing the containment by two for arrayed-definers. They are respectively named tetrational, pentational and so on.

Hyper-Condensed Arrays

(a{b}c) = A (c, c, c…, c) or b-length array of c-valued entries-level condensed array with a (c, c…, c)-length series of (c, c…, c)-length definers applied to (c, c…, c)-length arrays with a-valued entries

If two or more definers are used within the curly brackets, the full array (excluding a) is used for length and entries, and the containment will increase by one

Condensing levels are ordered with linear as 1, planar as 2, and so on

For multiple bracket sets, the containment is multiplied by however many brackets are used

Black Hole Arrays

(a/b\c) uses the same rules in relation to hyper-condensed arrays as the hyper-condensed arrays use in relation to the condensed arrays, except that the effect is multiplied by 10 for the containment rules, and c^b is used for length and entries for single-definers

All forms of this notation can be combined together, using their respective rules to solve

The final function for this notation is EPAN(n), or F(n), which means that you have to apply n to every variable possible within this notation. You can also use arrays within each function, whether or not it follows my system or another system.