User blog:GamesFan2000/GamesFan's Step Function

This is a step function. A couple of functions of this type have shown up on the wiki blog page. Let’s set some rules.

I’ll represent the function as S(n).

Rules
Rule 1: If the last entry is zero, it is removed. This takes up one step.

Rule 2: If the last entry is a positive integer larger than zero, perform the following procedures:

1. Take the result of (n+1)^s, where s is the step in which the decomposition happens on.

2. Decompose n into (n+1)^s n-1’s. If n=1, this is all that is needed.

3. If n>1, follow each n-1 with (n+1)^s n-2’s, follow each n-2 with (n+1)^s n-3’s, and so on until you follow each n-(n-1) with (n+1)^s n-n’s.

All of the above encompasses one step, the step from which s is derived.

Examples
S(0) and S(1) are trivial. S(0) decomposes into an empty set, so it equals 2. S(1) will decompose into S(0, 0, 0, 0), so S(1) equals 6.

S(2) is where things get dicey.

1. (2)

2. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹) (The superscript refers to how many zeroes there are in between each 1)

...

11. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1)

12. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 04105)

...

4117. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1)

4118. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1, 0, ...(2^4118)+9 0’s...0)

...

(2^4118)+4127. (1, 09, 1, 09, 1, 09, 1, 0⁹, 1, 0⁹, 1, 0⁹, 1) (Yeah, good luck with that one.)

S(2) is unspeakably huge. It’s clearly much larger than a googolplex, and I mean MUCH larger, if the last step I showed didn’t already indicate it. And it’s just the third number we can derive from the sequence. S(3) would dwarf S(2), for obvious reasons.

Inspiration and Shoutout
I was mainly inspired by PlantStarAlpineer’s Cobra function. The Cobra function is also very powerful, but my function dominates it. Even so, shoutouts to PlantStar for the concept.