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In mathematics, hexation (or hyper-5) is the next hyperoperation (infinite sequence of arithmetic operations) after Pentation and before Heptation. It is defined as iterated (repeated) pentation

${\textstyle a\uparrow \uparrow \uparrow \uparrow b=\underbrace {a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (\cdots (a\uparrow \uparrow \uparrow a)))} _{b{\text{ copies of }}a}}$

Notation

There is little consensus on the notation for hexation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others.

• Pentation can be written as a hyperoperation as ${\displaystyle a{b}}$
• In Knuth's up-arrow notation, hexation can be represented as ${\displaystyle a\uparrow \uparrow \uparrow \uparrow \uparrow b}$ or ${\displaystyle a{\uparrow }^{5}{b}}$