A244382 Decimal expansion of the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk.

7, 1, 8, 7, 9, 0, 3, 3, 5, 1, 6, 4, 1, 0, 6, 2, 2, 9, 4, 4, 0, 5, 1, 1, 7, 5, 4, 9, 2, 4, 4, 4, 2, 1, 0, 6, 7, 5, 4, 5, 7, 8, 4, 1, 8, 5, 4, 1, 5, 4, 2, 8, 7, 5, 4, 9, 5, 8, 0, 6, 6, 6, 3, 7, 2, 8, 2, 0, 0, 5, 2, 6, 6, 4, 4, 0, 0, 9, 4, 0, 6, 7, 4, 3, 4, 9, 5, 0, 8, 8, 5, 5, 8, 5, 3, 8, 8, 2, 7, 4, 8

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.4 John Constant, p. 466.

Links

Table of n, a(n) for n=1..101.

Julian Gevirtz, An upper bound for the John constant.

Formula

exp(lambda*Pi), where lambda is the positive solution of the equation Pi/(exp(2*Pi*lambda)-1) = Sum_{k > 0} k/(k^2+lambda^2)*exp(-k*(Pi/(2*lambda))).

Example

7.187903351641062294405117549244421...

Mathematica

eq = Pi/(Exp[2*Pi*x] - 1) == Sum[(k/(k^2 + x^2))*Exp[-k*(Pi/(2*x))], {k, 1, Infinity}]; lambda = x /. FindRoot[eq, {x, 1/2}, WorkingPrecision -> 102] // Re; RealDigits[Exp[lambda*Pi]] // First
RealDigits[N[E^(Pi Root[{(E^(2 Pi #) - 1) Beta[E^(-Pi/(2 #)), 1 - I #, -1] + (E^(2 Pi #) - 1) Beta[ E^(-Pi/(2 #)), 1 + I #, -1] + 2 Pi # &, 0.6278342676872}]), 100] // Chop][[1]] // Most (* Eric W. Weisstein, Dec 08 2017 *)

Crossrefs

Cf. A244381 (lambda).

Sequence in context: A199439 A153625 A011100 * A111293 A019661 A200130

Adjacent sequences: A244379 A244380 A244381 * A244383 A244384 A244385

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jun 27 2014

Status

approved