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The paper attached is a short paper about nested tetration.
The hyperoperator hierarchy continues the operations of addition, multiplication and exponentiation with tetration, pentation and so on. Tetration, or hyper(4), is the 4th operator in the hyperoperator hierarchy. Pentation, or hyper(5), is the next hyperoperator after tetration. In other words, pentation is repeated tetration, in the same way that tetration is repeated exponentiation.
For example: 3^^3=3^(3^3), 4^^4=4^(4^(4^4)) and 4^^^4=4^^(4^^(4^^4))
So we see that hyper(n) uses top-down bracketing as this is what gives a new operator.
However, hyper(4) can use a bottom-up [bu] bracketing style as well, for example:
(3^^3)[bu]= (3^3)^3 = 3^(3^2)
In general, every hyper(n) could use a bottom-up [bu] bracketing method.
Nested tetration is a fractal style of tetration: [(…).(…).(…)].[(…).(…).(…)].[(…).(…).(…)]
This nested pattern can be continued indefinitely, with as many nestings, and nestings of nestings as desired. It is actually the same same as pentation using the bottom-up bracketing style.
I thought about using the term “clustered” tetration instead of “nested” tetration, but I think that “nested” describes the fractaline process more clearly. Also, “clustered tetration” could be used to describe top-down exponential bracketing of normal tetration. In other words, clustered tetration more naturally refers to tetration of standard tetration numbers:
(n^^m)^((n^^m)^((n^^m)^(n^^m))) …
So clustered tetration is not fractaline whereas nested tetration is fractaline.
There is only one situation where clustered tetration coincides with nested tetration:
(n^^n)^^n=(n^^n)^((n^^n)^((n^^n)^…^((n^^n)^((n^^n)^(n^^n))…) with n occurences of (n^^n)
In my paper “Nested tetration” I define a flexible and extensible notation for a new recurrence relation that describes the phenomena of nested tetration.