(I made a mistake, 217940004309743 != 217940004309744 . . . My laziness, I apologize! /-: ) But I think I found a better answer to your question. See members.bex.net/jtcullen515/math7.htm (Jim Cullen, "An approximation of pi from Monster Group symmetries"). He starts with s^3 + 744, then proposes successively better approximations to e^(pi sqrt(163)) : s^3 + 744 - 196884/s^3 s^3 + 744 - 196884/s^3 + 167975456/s^6 s^3 + 744 - 196884/s^3 + 167975456/s^6 - 180592706130/s^9 then states that the next term is "barely identifiable as the integer 217940004309743" and suggests that further terms do not add to the elegance (of extending the near-exact approximation). He then goes on to that other series with the 217940004309744 term. So I guess no, those coefficients are probably not definitive, but still if they're in a paper somewhere they might as well also be in OEIS. - Robert On Wed, Dec 22, 2010 at 00:09, Robert Munafo <mrob27@gmail.com> wrote:

[...] (He and Jejjala, "Modular Matrix Models", arXiv:hep-th/0307293v2 , page 20 ) [...] mathoverflow.net/questions/4775/why-are-powers-of-exppisqrt163-almost-integers

[...]

On Tue, Dec 21, 2010 at 21:38, Bill Gosper <billgosper@gmail.com> wrote:

...

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

-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com