Mizugadro, pentation, Book

[sheldonison]

Was there anything of interest to you in [url=http://math.eretrandre.org/tetrationforum/showthread.php?

No. Sorry. My Mathematica fails to run the code suggested,
<< Tetration`
NaturalIterate[Series[Tetrate[E, x], {x, 0, 3}], z]

mode=linear&tid=372]Andy's 2009 pentation thread[/url]? My first post to the thread includes pari-gp code, post#10, Oct30th 2010 and gives the fixed points and Taylor series for Pentation to higher precision than your results

? How can I see that the precision is better?
Have you calculated the residual at the substitution into the transfer equation?
Or comparison with the explicitly–holomorphic Taylor expansion?

There is a Taylor series in that post#10, for Pent(x-1); you could compare it to results you have. When I rerun in higher precision today, the Taylor series coefficients I first posted are accurate to approximately 21 decimal digits; that Pentation result used a 32 decimal digits Tetration implementation for its base. The pentation pari-gp program hasn't been updated in over three years ... I would probably clean it up if I were writing it today, and also write more math equations (instead of just pari-gp code). The most recent version I have (roughly the same as post#13) works fine, and can be run in arbitrary precision and gives pent(-0.5)=0.4910543386356481974128179471452718984517, which gives an error term vs that very first post of pent(-0.5)=-7E-22.