Pentation can be written in Knuth's up-arrow notation as or . In this notation, represents the exponentiation function , which may be interpreted as the result of repeatedly applying the function , for repetitions, starting from the number 1. Analogously, , tetration, represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. And the pentation represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. Alternatively, in Conway chained arrow notation, . Until a consensus for the definition of tetration to real heights is reached, pentation can't even be defined to an integer exponent higher than 1 when it's base is not a whole number.
The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if is defined by the Ackermann recurrence with the initial conditions and , then .
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