A way to define Graham's Number and Little Graham (Graham-Rothschild Number) in a more elegant way, using an array notation instead of using ugly-looking functions. This was originally in the form of a blog post on Googology Wiki.
Rules
All entries in all arrays using this notation must be nonzero positive integers. An example of a valid array is [3,4,5,6,7].
@ is the rest of the array and can only contain numbers, {}'s (if we go to the extension) and commas (separators).
If an array has 3 or less entries, the rules are exactly the same as 3-entry arrays in BEAF/Bird's array notation.
If an array has 4 or more entries, we set forth the 3 following rules:
Rule 1: [a,1@] = a
Rule 2: [@1] = [@]
Rule 3: when b > 1, [@a,b] = [@[@a,b-1]]
Extension
An additional rule: if anything in {} brackets terminates an array, it degenerates into ",1". E.g. [10,10,10{1}1] = [10,10,10{1}] (because rule 2) = [10,10,10,1] = [10,10,10].
We define [a,b{1}2] as [a,a,...,a] with b a's.
Applying rule 3, we can also define [a,b,c{1}2] as [a,[a,b,c-1{1}2]{1}2].
We can continue and arrive at [a,b{1}3] which is [a,a,...,a{1}2] with b a's before the {1}.
[a,b{1}c] = [a,a,...,a{1}c-1] with b a's before the {1}.
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Examples
For Graham's Number
G(1) = [3,3,4]
G(2) = [3,3,4,2] = [3,3,[3,3,4,1]] = [3,3,[3,3,4]]
G(64) = [3,3,4,64] = [3,3,[3,3,4,63]] = [3,3,[3,3,[3,3,4,62]]] and so on.
G^{G}(64) = [3,3,4,64,[3,3,4,64]] = [3,3,4,64,1,2] = [3,3,4,64,[3,3,4,64,1,1]] (we remove tailing 1's)
For Little Graham:
g(1) = [2,3,12]
g(7) = [2,3,12,7] = [2,3,[2,3,12,6]] and so on.
Comparison with Fast-growing hierarchy
Ordinals are shown as the "order type"
[a,b,c] ~ w
[a,b,c,d] ~ w+1
[a,b,c,d,e] ~ w+2
[a,b,c,d,e,f] ~ w+3
[a,b{1}2] ~ w2
[a,b,c{1}2] ~ w2+1
[a,b,c,d{1}2] ~ w2+2
[a,b{1}3] ~ w3
[a,b,c{1}3] ~ w3+1
[a,b{1}4] ~ w4
[a,b{1}c] ~ w^2
Therefore, the limit of the well-defined part of this notation is w^2 (or around the limit of BEAF 4-entry arrays).
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