08/01/2014, 11:57 PM

(08/01/2014, 11:36 PM)sheldonison Wrote: Jay,

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.

I'm working on a set of generator polynomials that would allow me to compute an arbitrary term in the sequence in log-polynomial time. For instance, the naive approach requires O(n) calculations to calculate the nth term.

The generator polynomials are a set of log_2(n) polynomials of degree 1 through log_2(n). Calculating the kth polynomial requires calculating k+1 points from the (k-1)th polynomial. I haven't fully worked out the math, but I think it runs in something like O(log_2(n) ^ m) time, where m is a small constant, maybe 3 or 4. So, for example, calculating the 2^100th term in the sequence took about 10 minutes (as opposed to O(2^100) calculations, which would take forever on pretty much any processor).

I'll try to post gp code next week (it's in SAGE at the moment, and I'm not sure who else is using SAGE besides me... Andy and Henryk were using it a few years ago, not sure if they still do...)

~ Jay Daniel Fox