R50: My first actual googolism! It grows extremely similar to Graham's number, in the sense that it has a starting iteration that then exponentially grows the number as it goes on.
Here's the rules for iterations of R using a modified version of Bowers' array notation:
If n=1:
Rn = (500, [500], 500)
If n≥ 2:
Rn = (500, [500, {Rn-1}, 500], Rn-1)
Simply keep using this formula until you reach R50, like G64.
Shemaiah's Array Notation: First notation! Subject to change, as it's my first notation. Separator is /. d is the amount of dimensions in a given array.
Rules are as follows:
[a]=a+1
[a, b] = a^b
[a, b, c] = a^(b)c (a^c with b number of arrows)
[a, b, c, d] = (a^(b)c)^d
[a, b, c]/[d, e, f] = [a, b, c]^^[d, e, f]
[a, b, c]₃/[d, e, f] = [a, b, c]₂[a, b, c]₂[a, b]/[d, e, f]
[a, b, c]//[d, e, f] = [a, b, c]^^^^[d, e, f]
In English:
If there is only one argument, it is the original number plus one.
If there are only two arguments, exponentiate a by b.
If there are three arguments (No separators), a is exponentiated by c with b number of arrows.
When there are multiple dimensions of the array with a separator in between, turn the separator into d amount of up arrows (dimensions of the array are exponentiated by each other).
If there are more than three entries, simply exponentiate the entire row without the last entry by the last entry. If there are five or more entries, exponentiate the last number in parentheses by the last number outside parentheses. Repeat this process as many times as needed (basically, if you need to, just make power towers on the last number in parentheses).
If a subscript number appears next to a dimension, split it into that number of copies then place the number on each copy and subtract one. If the number is 2 or greater, keep splitting until you get rid of the number. Do not add the number to the last entry of the side that is being copied. When you are done, remove the last entry from the side that was copied.
If there are multiple separators, multiply the number of separators by d, then apply to all separators separating dimensions.