hyperoperation

In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written using n − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by: a[n]b = \underbrace{a[n-1](a[n-1](a[n-1](\cdots a[n-1](a[n-1](a[n-1]a))\cdots)))}_{\displaystyle b \mbox{ copies of } a},\quad n \ge 2

It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function: a[n]b = a[n-1]\left(a[n]\left(b - 1\right)\right),\quad n \ge 1