08/01/2014, 11:36 PM
(This post was last modified: 08/01/2014, 11:49 PM by sheldonison.)

(08/01/2014, 01:53 AM)jaydfox Wrote: If we take k very high, the continuous function seems to converge on a constant multiple of the discrete sequence... Based on analysis of the first 2^18 terms in the sequence, the constant is somewhere between 1.083035 and 1.083089, and probably pretty close to the center of that interval, about 1.083062.Jay,

....

I did some googling, and I found this reference:

http://link.springer.com/article/10.1007/BF01933448

...

So I'm hoping that if someone could get access to that article, they might be able to shed a little light on this constant factor.

I can't read the article either; but I am very intrigued by this slowly converging ratio, and by your Taylor series approximation! Very impressive. I'm still working on understanding your "log_2(n)" term count memory model for the function too, but the Taylor series approximation is easy to calculate for reasonably large inputs.

At some point, I'd like to see if I can figure out how fast grows, as compared to tetration, . Of course, a graph of the Taylor series in the complex plane would be fun too, as well as the location of the zeros.

- Sheldon