Project with the sequences (only limited to expanding arrays an amount of times):

φ(ω,0) In Snap!

After a week, I found out there is just one simple way to achieve $$\varphi(\omega,0)$$. I originally made another sequence system related to this (had same growth rate but more complicated) and someone said it was the same as the LPrSS, which i hadn't heard of. Then I checked it out and it was way simpler. It uses systems called **Difference Sequences**, which are duplicates of a sequences with each duplicate being increased by a difference between 2 items. The new system is below:

It might look complicated, but in reality, it's not! It's just a little complicated to implement.

Examples:

The block (LPrSS (0)(2) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_0}(n)$$

The block (LPrSS (0)(2)(1)(3) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_0^2}(n)$$

The block (LPrSS (0)(2)(1)(3)(2) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_0^\omega}(n)$$

The block (LPrSS (0)(2)(1)(3)(2)(4) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_0^{\varepsilon_0}}(n)$$

The block (LPrSS (0)(2)(2) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_1}(n)$$

The block (LPrSS (0)(2)(3) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_\omega}(n)$$

The block (LPrSS (0)(2)(4) ⏴ ⋮⏵ (n)) returns $$f_{\varepsilon_{\varepsilon_0}}(n)$$

The block (LPrSS (0)(3) ⏴ ⋮⏵ (n)) returns $$f_{\zeta_0}(n)$$

The block (LPrSS (0)(4) ⏴ ⋮⏵ (n)) returns $$f_{\varphi(3,0)}(n)$$

The block (LPrSS (0)(5) ⏴ ⋮⏵ (n)) returns $$f_{\varphi(4,0)}(n)$$

The block (LPrSS (0)(m) ⏴ ⋮⏵ (n)) returns $$f_{\varphi(m-1,0)}(n)$$

**The block (LPrSS (0)(n) ⏴ ⋮⏵ (n)) returns $$f_{\varphi(\omega,0)}(n)$$!!!**

Meaning after all the months i've been on the forums, I've finally made (or implemented) a function that creates numbers up to $$f_{\varphi(\omega,0)}(n)$$. In my opinion, Linear and Planar BEAF were

**harder**to implement than this, even though their growth rates are on$$f_{\omega^\omega}(n)$$ ly $$f_{\omega^{\omega^2}}(n)$$, which are (LPrSS (0)(1)(2) ⏴ ⋮⏵ (n)) and (LPrSS (0)(1)(2)(2) ⏴ ⋮⏵ (n)) in the new blocks. $$\omega^\omega$$, is nothing compared to $$\varepsilon_0$$ (or 0,2), which is nothing compared to $$\varepsilon_1$$ (or 0,2,2), which is nothing compared to $$\zeta_0$$ (or 0,3), which is nothing compared to the legendary $$\varphi(\omega,0)$$ (or 0,n), which is the limit of this system of sequences.