User blog:GamesFan2000/Front-To-Back Array Notation (FTBAN)

Front-To-Back Array Notation is a system of arrays that starts from the front and uses some clever rules to build some absolutely massive numbers. There are some very important terms to understand before going in to the meat of the notation.

Terminology

Pilot(n): The first non-one entry of the array. Can be any entry.

Co-Pilot(a): The entry directly after the pilot.

Vacancies: Any ones after the pilot are called vacancies.

Outsiders: Any ones before the pilot are called outsiders.

Tail(b): The final entry of the array.

Rules

All single-entry arrays, (n), default to n.

For two-entry arrays, (n, a), the following rules are to be used:

If both n and a are equal to 1, the array defaults to 1.

If the above doesn’t apply, but a=1, go through the following logic:

Replace a with n and reduce n by 1. (4, 1)=(3, 4) Create an n+2 level operation on a and reduce n by 1. (3, 4)=(2, 4^^^4)=(1, (4^^^4)^^(4^^^4)) Once n=1, replace it with the expression you just built, and you’re done.

If neither of the above apply, then go through the following logic:

Create an n+2 level operation on a and reduce n by 1. (5, 5)=(4, 5^^^^^5) Repeat this until n equals 1. (4, 5^^^^^5)=(3, (5^^^^^5)^^^^(5^^^^^5))=(2, ((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5)))=(1, (((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5)))^^(((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5)))) Once n=1, replace n with the new expression, and replace the expression with the original a. Reduce a by 1. (1, (((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5)))^^(((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5))))=((((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5)))^^(((5^^^^^5)^^^^(5^^^^^5))^^^((5^^^^^5)^^^^(5^^^^^5))), 4). Repeat the process until a=1, and then refer to the rules for that situation.

For three-entries and longer, (n, a, …b):

If all of the entries are 1, then the array defaults to 1.

If the above doesn’t apply, then the rules for an entry equalling 1 and no entries equalling 1 apply, with 1 modification: (1, 3, 3)=(1, 2, 3^^^3), except if the original pilot isn’t reduced to 1, in which case the original rules apply: (2, 3, 3)=(1, 3^^3, 3), except if the array started with the first entries as outsiders, for which the modification trumps the original rule. For any entry, once the previous entries are all equal to 1, replace all of them with the new expression and replace the current entry with whatever you originally had and reduce that original entry by 1: (1, 1, 1, …3^^3, …)=(3^^3, 3^^3, 3^^3, …2, …). For the tail, if whatever it originally was is reduced to 1, and you’ve completed the process, then replace every previous entry with the new expression and crop the tail: (1, 1, 1, 1, …3^^3)=(3^^3, 3^^3, 3^^3, 3^^3, …). These rules apply for all arrays of at least three entries or longer. Note that once you replace the outsiders, their process essentially resets, so you’ll recommence the reductions of original entries. FTBAN has extreme recursion sequences such that you wouldn’t be able to fully unpack it into standard form most of the time. Growth rates are appreciated.