A178449 Conjectured expansion of exp(Pi*sqrt(163)) in powers of t, where t = 1/(640320)^3.

1, 744, -196884, 167975456, -180592706130, 217940004309743, -19517553165954887, 74085136650518742, -131326907606533204

Offset

-1,2

Comments

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.

References

R. W. Gosper, Posting to the Math Fun Mailing List, Dec 21 2010

Links

Table of n, a(n) for n=-1..7.

Jim Cullen, An approximation of Pi from Monster Group symmetries

Example

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.

Prog

/* GNU bc code, computes a(0) through a(7) */
define trunc(x) { auto sc, t; sc=scale; scale=0; t=x/1; scale=sc; return(t) }
scale = 200; pi = 4 * a(1); r = e(pi * sqrt(163)); s = 640320;
c0 =  1 + trunc(r - s^3);
c1 = -1 - trunc(((s^3 + c0) - r) * s^3);
c2 =  1 + trunc((r - (s^3 + c0 + c1/s^3)) * s^6);
c3 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6) - r) * s^9);
c4 =  1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9)) * s^12);
c5 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12) - r) * s^15);
c6 =  1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12 + c5/s^15)) * s^18);
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);

Crossrefs

Cf. A000521, A091406, A178451, A066396.

Sequence in context: A192731 A288261 A000521 * A178451 A066395 A161557

Adjacent sequences: A178446 A178447 A178448 * A178450 A178451 A178452

Keyword

sign,more

Author

N. J. A. Sloane, Dec 22 2010, based on a posting by R. W. Gosper to the Sequence Fans Mailing List, Dec 21 2010

Extensions

Cullen link, bc code, and a(8) from Robert Munafo, Dec 23 2010

Status

approved