[seqfan] Re: e^(pi rt 163) series suggested by Bill Gosper (OEIS A178449)
apovolot at gmail.com
Fri Dec 24 20:57:17 CET 2010
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
in case of 3)
Above results gives me hope that Gosper's expansion coefficients are
some what applicable to all considered by me exp(Pi*sqrt(19+24*n)
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
> 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
> s1^3 + 744 - 196884/s1^3 + 167975456/s1^6 - 180592706130/s1^9 +
> 217940004309743/s1^12-19517553165954887/s1^15+74085136650518742/s1^18 -...
> = 147198695.9986624619692069551 - ...
> exp(Pi*sqrt(19+24*2)) = 147197952743.9999986624542245
> For the case 2)
> s2 = (24*40/10) = 96
> 217940004309743/s2^12-19517553165954887/s2^15+74085136650518742/s2^18 -...
> = 885479.7776801543199810507716 - ...
> 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
>> I looked around the web for the numbers (167975456 etc.); NJAS created
>> 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
>> It seems I found a definition that produces those numbers, though maybe
>> the intended definition?
>> Feel free to suggest something better (like more concise code perhaps :-)
>> edit/comment on A178449.
>> - Robert
>> Robert Munafo -- mrob.com
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