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

Alexander P-sky apovolot at gmail.com
Fri Dec 24 07:53:21 CET 2010

```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

```