User blog:GamesFan2000/Binary Numeral Array Function

In mathematics, a binary numeral is a base-2 expression represented by 0’s and 1’s. This is how computers calculate their logic. The number of states that a binary system can be in is 2n, where n is the number of binary digits. For the purposes of this function, we’ll use Bird’s array notation. Anything that is defined in BAN is fair game for this function. This is the Binary Numeral Array Function, or BNAF.

Definition

This function is represented by the general expression B(n), where n is the number of binary digits. Here’s the rules:

The number of variables/entries in the array can be no more than the number of possible states in an n-bit expression, and the number of symbols used that aren’t entries or variables can be no more than the number of possible states in an n-bit expression. You are allowed to use arrays as entries/variables, but these also must conform with the rules of the main array. The maximum number of ‘containment levels’ allowed, where {{a, b}, {a, b}} has a containment level of 1, is the number of possible states in an n-bit expression. A ‘state’ is defined as any base-2 expression in this context, i.e. 01, 1111, 0101100. Commas aren’t considered ‘symbols’ in this context.

Under the rules stated above, B(n) is the maximum possible value that can be achieved in BAN.