# [seqfan] Re: e^(pi rt 163) series suggested by Bill Gosper (OEIS A178449)

Alexander P-sky 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

and

in case of 3)

147197952744/147198695.9986624619692069551= 999.994951

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)
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
>

```