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%I #14 Dec 13 2024 09:41:27
%S 3,5,1,3,8,3,0,7,1,9,1,2,5,1,6,1,2,0,6,2,0,7,8,3,7,0,9,3,2,3,8,8,2,3,
%T 5,8,7,1,0,9,1,3,4,2,1,1,9,5,1,2,8,4,3,6,8,1,8,2,5,4,1,8,5,2,5,3,4,9,
%U 2,1,8,6,0,8,7,7,3,5,3,0,6,2,2,4,5,1,3,9,8,4,8,8,7,6,5,9,9,9,7,5,7,3,9,5
%N Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace's limit constant A033259.
%C The boundary value problem y''(x) + c*exp(y(x)) = 0, y(0) = y(1) = 0 and c > 0, has 0, 1 or 2 solutions when c > gamma, c = gamma and c < gamma, respectively. [After Steven Finch]
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.8 Laplace limit constant, p. 266.
%H G. C. Greubel, <a href="/A248916/b248916.txt">Table of n, a(n) for n = 1..20000</a>
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 32.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a>
%e 3.5138307191251612062078370932388235871...
%t digits = 104; lambda = x /. FindRoot[x Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> digits + 5]; gamma = 8*lambda^2; RealDigits[gamma, 10, digits] // First
%Y Cf. A033259, A085984.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Oct 16 2014