[seqfan] Re: e^(pi rt 163) series suggested by Bill Gosper (OEIS A178449)
Alexander P-sky
apovolot at gmail.com
Sat Dec 25 15:18:47 CET 2010
Dear Robert,
> Even if you eliminate the divide by 10 thing, there is nothing special.
Well, who knows - see my point below ...
> 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.
Above may just mean that for the smaller constants exp(pi*sqrt(67)),
exp(pi*sqrt(43)) and exp(pi*sqrt(19)) - due to the smaller value of
the "s" bases
- the conversion may just go much slower for those
> That means that for those smaller cases, a different series of coefficients
> would be needed.
It may might just mean, contrary to Robert's above conclusion, that
more additional (not others !) coefficients are needed - since as it
is defined by Gosper's expansion formula, the sign alternating
expansion has the infinite number of coefficients which indeed need to
be found.
Regards,
Alexander R. Povolotsky
===================================================================
On 12/25/10, Robert Munafo <mrob27 at gmail.com> wrote:
> 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:
---------- Forwarded message ----------
From: Alexander P-sky <apovolot at gmail.com>
Date: Fri, 24 Dec 2010 14:57:17 -0500
Subject: Re: [seqfan] e^(pi rt 163) series suggested by Bill Gosper
(OEIS A178449)
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: Bill Gosper <billgosper at gmail.com>, "N. J. A. Sloane"
<njas at research.att.com>, Robert Munafo <mrob27 at gmail.com>,
rcs at xmission.com, Bill Gosper <rwg at sdf.lonestar.org>, "Mark A. Thomas"
<monstrousgaugetheory at gmail.com>
Perhaps I may need explicitly point to the fact (which may be was not
so obviously clear seen in my previous email) that:
in case of 2)
884 736 744 / 885 479.7776801543199810507716 = 999.160869
and
in case of 3)
147197952744/147198695.9986624619692069551= 999.994951
Above results give me hope that Gosper's expansion coefficients are
some what applicable to all considered by me exp(Pi*sqrt(19+24*n)
cases
Alexander R. Povolotsky
====================================
On 12/24/10, 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
> for the four cases below:
> 1) n=0, k= 4
> 2) n=1, k= 40
> 3) n=2, k= 220
> 4) n=6, k = 26680
>
> Now let us look at the power base of first member on the right hand
> side in the above approximation being divided by 10, that is (24*k/10)
>
> For the last case (that is 4) above)
> (24*26680/10) = 640320, which is Gosper's "s"
>
> 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
>
> For the case 2)
> s2 = (24*40/10) = 96
> and
> s2^3+744-196884/s2^3+167975456/s2^6-180592706130/s2^9+
> 217940004309743/s2^12-19517553165954887/s2^15+74085136650518742/s2^18 -...
> = 885479.7776801543199810507716 - ...
> while
> exp(Pi*sqrt(19+24*1)) = 884736743.9997774660349066619
>
> Alexander R. Povolotsky
> ====================================
> On 12/23/10, Robert Munafo <mrob27 at gmail.com> wrote:
>> (Moving discussion over to seqfan)
>>
>> A178449 are coefficients for a series sum to "Ramanujan's constant" e^(pi
>> *
>> sqrt(163)), suggested over in math-fun by Bill Gosper:
>>
>> Subject: e^(pi rt 163) =
>> s^3 + 744 - 196884/s^3 + 167975456/s^6 - 180592706130/s^9 +
>> 217940004309743/s^12 - 19517553165954887/s^15 + 74085136650518742/s^18 -
>>> ...
>>
>> where s = 640320. Only the 1st three terms are in EIS. Are the rest
>> well
>>> defined?
>> --rwg
>>
>>
>> I looked around the web for the numbers (167975456 etc.); NJAS created
>> the
>> sequence.
>>
>> I added one link that I found, some code (using bc) that computes those
>> coefficients in a well-defined way, and the next term following the same
>> pattern.
>>
>> It seems I found a definition that produces those numbers, though maybe
>> not
>> the intended definition?
>>
>> Feel free to suggest something better (like more concise code perhaps :-)
>> or
>> edit/comment on A178449.
>>
>> http://oeis.org/draft/A178449
>>
>> - Robert
>>
>> --
>> Robert Munafo -- mrob.com
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