08/19/2014, 05:31 PM

(08/11/2014, 05:19 PM)jaydfox Wrote:I just found another esimate for a_1 (well, 1/a_1). Check this out:(08/11/2014, 04:51 PM)sheldonison Wrote: Nice post. What is the most accurate version of the scaling constant that you know of? Earlier, you posted, "1.083063".

I'm not having much luck finding references to the constant online. The closest two I've seen are the 1.083063 that I found earlier through a google search (post #3 in the current discussion), and a reference to 1/a_1 of 0.9233, on this page:

https://oeis.org/A002577

The latter has 4 sig-figs, though it's accurate to almost 5 (1/1.083063 ~= 0.923307)

http://www.sciencedirect.com/science/art...6503001237

Quote:Results on the number of binary partitions, Bn: In [8], Fröberg proves the following: Define

Then Bn=Cn·F(n), where (Cn) is a sequence bounded between 0.63722<Cn<1.920114 for all n⩾0. It is estimated (but unproven) that Cn tends to a limit C:≈0.923307±0.000001 as n→∞.

[8] C.-E. Fröberg

Accurate estimation of the number of binary partitions

BIT, 17 (1977), pp. 386–391

Abstract

Many authors have worked with the problem of binary partitions, but all estimates for the total number obtained so far are restricted to the exponential part only and hence very crude. The present paper is intended to give a final solution of the whole problem. © 1977 BIT Foundations.

And there it is: C:≈0.923307±0.000001, where C is 1/a_1 as I defined it.

That abstract looks familiar. I'm pretty sure it's the same paywalled paper I linked to before. It was written the year I was born, so it just seems a shame that there's still such a hefty fee to access it (over $40).

~ Jay Daniel Fox