My favorite integer sequence

[tommy1729]

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.
....
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.

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 \( f^{o n}(1) \) grows, as compared to tetration, \( \exp^{o n}(1) \). Of course, a graph of the Taylor series in the complex plane would be fun too, as well as the location of the zeros.

\( f^{o n}(x) \) grows like \( exp(a^{2^n - 1} ln(x)^{2^n} + C) \).

Much much slower then exp or even its half-iterate.
And that for a function that is similarly defined. (second derivative = Original => second difference = Original )
Thats what I meant by counterintuitive.

I noticed the resemblance of the plots with the fake half-iterate exp too.

For the zero's Im convinced they are all real.

Math can be deceptively complicated or easy and its not even clear which one it is or which occurs most often.

It reminds me of the student who said at the beginning of calculus :

I know what a function is !!

and ended : I dont know what a function is !!

---

One of these generalizations is the following :

f(n) = f(n/2) + f(n/3) + ... + f(sqrt(n))

where all divisions and sqrt are rounded.

Mick posted an ("the" ??) analytic analogue here :

http://math.stackexchange.com/questions/...econd-type

Leibniz rule seems to get into circular stuff at first sight.

Thue-Morse ideas is another thing I discussed with him resulting in this :

http://math.stackexchange.com/questions/...nd-new-grh

Those darn zero positions

regards

tommy1729