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A244382 Decimal expansion of the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 27 2014

STATUS

approved

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Last modified October 20 08:05 EDT 2019. Contains 328252 sequences. (Running on oeis4.)