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%I #25 Jul 27 2021 05:04:37
%S 1,744,-196884,167975456,-180592706130,217940004309743,
%T -19517553165954887,74085136650518742,-131326907606533204
%N Conjectured expansion of exp(Pi*sqrt(163)) in powers of t, where t = 1/(640320)^3.
%C R. W. Gosper asks if the coefficients are well-defined. Until this is answered, the sequence is only conjectural. This sequence is very close to A178451, but presumably different from it.
%D R. W. Gosper, Posting to the Math Fun Mailing List, Dec 21 2010
%H Jim Cullen, <a href="https://web.archive.org/web/20210421094535/http://members.bex.net/jtcullen515/math7.htm">An approximation of Pi from Monster Group symmetries</a>
%e e^(Pi*sqrt(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. Now set t = 1/s^3.
%o /* GNU bc code, computes a(0) through a(7) */
%o define trunc(x) { auto sc,t; sc=scale; scale=0; t=x/1; scale=sc; return(t) }
%o scale = 200; pi = 4 * a(1); r = e(pi * sqrt(163)); s = 640320;
%o c0 = 1 + trunc(r - s^3);
%o c1 = -1 - trunc(((s^3 + c0) - r) * s^3);
%o c2 = 1 + trunc((r - (s^3 + c0 + c1/s^3)) * s^6);
%o c3 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6) - r) * s^9);
%o c4 = 1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9)) * s^12);
%o c5 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12) - r) * s^15);
%o c6 = 1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12 + c5/s^15)) * s^18);
%o c7 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12 + c5/s^15 + c6/s^18) - r) * s^21);
%Y Cf. A000521, A091406, A178451, A066396.
%K sign,more
%O -1,2
%A _N. J. A. Sloane_, Dec 22 2010, based on a posting by R. W. Gosper to the Sequence Fans Mailing List, Dec 21 2010
%E Cullen link, bc code, and a(8) from _Robert Munafo_, Dec 23 2010
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