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%I #11 Nov 08 2015 03:57:20
%S 1,5,4,3,4,0,4,6,3,8,4,1,8,2,0,8,4,4,7,9,5,8,7,0,9,7,4,0,0,5,3,3,1,5,
%T 5,5,3,6,9,7,8,8,3,7,6,4,7,1,9,2,6,2,6,9,4,5,9,7,2,4,1,3,6,2,4,8,9,8,
%U 8,7,7,3,9,6,2,9,4,7,3,6,1,8,6,2,9,5
%N Decimal expansion of the solution of (x-1)/(x+1) = exp(-x).
%C Geometric interpretation: Consider the two curves y = exp(x) and its inverse y = log(x). They have two shared tangent lines, one with slope b, passing through the kissing points [a,b] and [1/b,-a], the other one with slope 1/b, passing through the kissing points [-a,1/b] and [b,a], where b = exp(a). The values of b and 1/b are reported in A263357.
%H Stanislav Sykora, <a href="/A263356/b263356.txt">Table of n, a(n) for n = 1..2000</a>
%F Equals log(A263357).
%e 1.543404638418208447958709740053315553697883764719262694597241362489...
%t RealDigits[x/.FindRoot[(x-1)/(x+1) == E^(-x), {x, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* _Vaclav Kotesovec_, Nov 06 2015 *)
%o (PARI) solve(x=0,10,(x-1)/(x+1)-exp(-x))
%Y Cf. A263357.
%K nonn,cons
%O 1,2
%A _Stanislav Sykora_, Oct 16 2015
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