[update:]

after I've written the following I got aware, that this should be better threaded under something like "polynomial tetration/pentation" since the basic powerseries for the tetration as I used it here was derived from the "polynomial tetration" using the diagonalization of the truncated real (square) Bellmatrix and its diagonalization - *not* of the triangular Bellmatrix of the regular tetration
While usually I handle the tetration having a series which has the height-parameter in the exponents of its single terms, I played a bit around with a powerseries-representation, and especially a powerseries-epansion for b^^h for b=4 and beginning at x=0 (instead of x=1 as usual). So that powerseries gives b^^0 = 0, b^^1=1,b^^2=4 and so on. Obviously that powerseries has no constant term. It begins like

PHP Code:

`b^^h= 0.938409102074*h - 0.167927302891*h^2 + 0.287554406742*h^3 - 0.151567019585*h^4 +... `

and for instance for b^^0.5 = 0.457214343478 .

Having a powerseries without constant term cries for iteration, so this allows to implement pentation to base 4 as an iteration of the current powerseries. Moreover, I have then a two-parametric formula (as usually in x and h in exp_b°h(x) if base b is fixed).

If I set x=1 and p="pentation-height"=1 I get the tetration beginning at 0 with height 1 (which gives 1) and this "iterated 1 time". If I iterate this 2 times, I get the following powerseries

PHP Code:

`((b^^h)^^^2 = exp_4°h(0))^^^2 = 0.880611642856*h - 0.305463247599*h^2 + 0.560396635817*h^3 - 0.482701855960*h^4 +... `

and for h=1 I evaluate this to the value 1. Surprisingly. That means, if I "iterate the tetration from 0 to 1" one time I get 1 and if I do this 2 times, I get still 1 and this comes out for all iterations/pentation-heights.

Hmm. Again, if I tetrate 0 to the height 1/2 I get

. This is done 1 time, so the p=1.

If I "do it two times" (???), p=2 I arrive at

PHP Code:

`(b^^0.5)^^^2 = 0.417108029167 `

. if I "do it 1/2 times" I get

PHP Code:

`(b^^0.5)^^^0.5 = 0.478298158357 `

Does someone have a pentation-implementation which could provide comparable/concurring values? Well, you may see: I'm not even feeling that I understood what that computation (and the change of that parameters) tells me conceptually at all...

Gottfried