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A244382
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Decimal expansion of the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk.
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1
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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
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OFFSET
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1,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.4 John Constant, p. 466.
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LINKS
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Table of n, a(n) for n=1..101.
Julian Gevirtz, An upper bound for the John constant.
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FORMULA
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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))).
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EXAMPLE
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7.187903351641062294405117549244421...
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A244381 (lambda).
Sequence in context: A199439 A153625 A011100 * A111293 A019661 A200130
Adjacent sequences: A244379 A244380 A244381 * A244383 A244384 A244385
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Jean-François Alcover, Jun 27 2014
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STATUS
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approved
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