tetrationlike

Tetrationlike is an informal term used in mathematics to describe objects, sequences, or functions that resemble tetration in either construction or growth. Tetration itself is iterated exponentiation, typically written as a ↑↑ n, representing a power tower of height n with a fixed base. Tetrationlike extends this idea to variants where the exponentiation is arranged in a tower but may involve changes such as varying bases at different levels, non-integer heights, or exponents defined by a rule rather than a single constant base.

In a basic tetrationlike sequence, one may define c1 as an initial value and set c_{n+1} = base_n^{c_n}, or more generally c_{n+1} = f_n(c_n) where f_n resembles exponentiation. In standard tetration, the base is constant and the height of the tower is an integer. Tetrationlike constructions also cover towers with level-dependent bases, or towers generated by iterated functions that mimic the exponential stacking, including varying rules for how the exponent and base relate at each level.

Growth behavior: Tetrationlike objects typically exhibit extremely rapid growth with height, often outpacing simple exponentials or even double exponentials. The precise growth depends on the defining rule, the sequence of bases, and whether heights are restricted to integers or extended to real or even complex values. Some discussions of tetrationlike ideas also consider infinite power towers and analytic continuations that extend the notion to broader heights or domains.

Examples: The classical tetration sequence with base 2 is a1 = 2, a_{n+1} = 2^{a_n}, yielding 2, 4, 16, 65536, and so on. A tetrationlike variant with varying bases, such as a1 = 3 and a_{n+1} = 2^{a_n}, produces 3, 8, 256, 2^{256}, etc.

See also: Tetration, power tower, hyperoperation, iterated exponentiation.