Double exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows much more quickly than an exponential function. For example, if a = b = 10:
Factorials grow more quickly than exponential functions, but much more slowly than doubly exponential functions. Tetration and the Ackermann function grow even faster. See Big O notation for a comparison of the rate of growth of various functions.
The inverse of the double exponential function is the double logarithm ln(ln(x)).
Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two. Integer sequences with this squaring behavior include
More generally, if the nth value of an integer sequence is proportional to a double exponential function of n, Ionaşcu and Stănică call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly exponential sequence plus a constant. Additional sequences of this type include
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