[seqfan] Re: e^(pi rt 163) series suggested by Bill Gosper (OEIS A178449)
Robert Munafo
mrob27 at gmail.com
Sat Dec 25 08:25:03 CET 2010
Alexander,
Your conjectures do not seem to play out. The divide by 10 thing doesn't
make sense at all, because the 744 term isn't being divided by 1000. These
will automatically be off by 0.999*744.
Even if you eliminate the divide by 10 thing, there is nothing special.
The following bc output shows that while the sequence [1] matches the
Ramanujan constant to within 1/s^18, for the smaller
constants exp(pi*sqrt(67)), exp(pi*sqrt(43)) and exp(pi*sqrt(19)) the sum
comes to within only (approximately) 1/s^10, 1/s^9 and 1/s^6 respectively.
That means that for those smaller cases, a different series of coefficients
would be needed.
In this bc output I set the precision to 200 digits beyond the decimal
point, set "pi" to 4*arctan(1), compute the Ramanujan-like constants, and
define the function "bg8(s)" to evaluate the sum of the 8 terms for a given
s (which was 640320 in [1]).
Then I compare bg8(s) to the corresponding Ramanujan constant and take the
logarithm to base s for each one, to show how far off it is. This should
give -18 corresponding to an error of 1/s^18 but that only happens in the r4
case.
Finally, I show an example of the "divide by 10" version you propose to show
that it's off by approximately 0.999 * 744 as one would expect.
$ bc -l
bc 1.06
Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type `warranty'.
scale = 200; pi = 4 * a(1);
r1 = e(pi*sqrt(19))
r2 = e(pi*sqrt(43))
r3 = e(pi*sqrt(67))
r4 = e(pi*sqrt(163))
define bg8(s) {
auto t;
t = s^3 + 744 - 196884/s^3 + 167975456/s^6 - 180592706130/s^9;
t = t + 217940004309743/s^12 - 19517553165954887/s^15;
t = t + 74085136650518742/s^18;
return(t)
}
# NOTE: I have edited the following by deleting all
# digits beyond the first 4 significant digits. -RPM
bg8(640320)-r4
.0000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000001527...
l(.)/l(640320)
-18.0518...
bg8(5280)-r3
.000000000000000000000000000000000000003799...
l(.)/l(5280)
-10.3207...
bg8(960)-r2
.0000000000000000000000000004843...
l(.)/l(960)
-9.1590...
bg8(96)-r1
.0000000000004835...
l(.)/l(96)
-6.2128...
bg8(528)-(r3/1000)
743.2546...
0.999 * 744
743.256
[1] (from Bill Gosper: s^3 + 744 - 196884/s^3 + 167975456/s^6 -
180592706130/s^9 + 217940004309743/s^12 - 19517553165954887/s^15 +
74085136650518742/s^18 - ...
> where s = 640320
On Fri, Dec 24, 2010 at 01:53, Alexander P-sky <apovolot at gmail.com> wrote:
> Let start with referring to the comment I made in A060295 referring to my
> observation that near integers in left hand side are the exact integers in
> the right hand side in the below approximation
> exp(Pi*sqrt(19+24*n) =~ (24*k)^3 + 31*24
> [...]
> By analogy for the case 3)
> s1 = (24*220/10) = 528
> and
> s1^3 + 744 - 196884/s1^3 + 167975456/s1^6 - 180592706130/s1^9 +
> 217940004309743/s1^12-19517553165954887/s1^15+74085136650518742/s1^18 -...
> = 147198695.9986624619692069551 - ...
> while
> exp(Pi*sqrt(19+24*2)) = 147197952743.9999986624542245
>
> [...]
--
Robert Munafo -- mrob.com
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