Basic R Function
Based onExponentiation
Growth rate\(f_{\omega}(n)\)

Here I bring you a series of powerful functions - R function. It stands "Recursion Ruler". The first one is basic R function. Just "basic", it's not so powerful.

It looks like nRm, where n and m are integers that n>0 and m>=0. R here is just a symbol, or an operator like plus(+), times(*), power(^), up-arrows(^^...^), Bowers' operator({{{2}}}), and so on. n, the number before R, is the base number. In my further extension, things after the R symbol will be many things, but the "nR" never change.


It's defined as follows:

Rule 1: nR0=10^n.         ----- base rule
Rule 2: nR(m+1)=((...(nRm)Rm...)Rm)Rm with n R's.

Basic R function solves from left to right automatically, so Rule 2 can be written as: nRm=nRm-1Rm-1...Rm-1 with n R's.

The base rule is not important, as it can be changed into anything when needed. We can change it into n+1, 2n, n^2, n^n, TREE(n), SCG(n), D(n)(Loader's function), or Rayo(n), etc. . But it doesn't matter. And, the growth rate showed in the function label is based on base rule is 10^n. What's important is how it grows from the base rule on.

More R functions

Brace notation

Linear array notation

Dimensional array notation

Nested array notation

Hyper nested array notation

To be continued...

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