Kerry, that is an *extremely* cool and thought-provoking set of images. I propose that the points on the boundary be called "Foias points", and then we can talk about the positive and negative real Foias points. Is anybody other than me slightly freaked out by the fact that the microscopic neighborhoods of the positive and negative real Foias points look so similar? They don't look similar at all when you look at the diagram as a whole. Note that the set of Foias points looks much more like a Julia set than it does like a Mandelbrot set, which makes sense because it is asking about the divergence of a *single* iterated function. I suspect that if we iterated (c + 1/f(n))^n (with an arbitrary constant c replacing 1 in the original), we would get a whole family of Julia sets plus a Mandelbrot analogue. On Sat, Sep 5, 2020 at 5:00 PM christopher landauer <topcycal@gmail.com> wrote:
hihi, all -
ok, i found part of the problem; it was my mistaken definition for a variable
i ran the same program with what i thought were going to be exponents n-1, n, and n+1 (and without GMP):
for n-1, it found the threshold in the interval
lo 1.187452351126496719
hi 1.187452351126496941
(this is the right answer for the original problem),
for n, it found the threshold in the interval
lo 1.842139054296648482
hi 1.842139054296648704
(this is what i reported before),
for n+1, it found the threshold in the interval
lo 2.380377436256564838
hi 2.380377436256565282,
and all of them exhibit the peculiar unique threshold property,
which leads me to wonder how far away from n the exponent can be and still have this threshold property
so these exponents are each off by 1 (what i named as ``n'' in the program was actually (n+1), as was correctly pointed out earlier by George Hart) -
oops
more soon,
chris
On 2020-09-04 11:15, George Hart wrote:
Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely.
Dan: Thanks for posting an interesting and surprising problem.
Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1)
All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice...
George http://georgehart.com
On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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