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%I #25 Jan 06 2024 18:56:46
%S 8,5,6,2,4,8,2,1,4,4,4,9,2,6,6,1,1,6,8,4,3,3,4,5,8,9,5,9,7,0,5,5,3,2,
%T 9,6,7,6,9,1,7,6,4,1,8,1,5,9,0,4,1,1,1,2,8,7,2,2,1,4,2,5,9,5,5,5,7,1,
%U 1,4,3,5,9,8,0,5,9,1,1,5,3,6,9,8,5,8,4,4,3,7,7,2
%N Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate of the point at which the line touches y = exp(-x).
%H V. G. Drinfel'd, <a href="https://doi.org/10.1007/BF01316982">A cyclic inequality</a>, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
%H Petros Hadjicostas, <a href="/A086277/a086277.pdf">Plot of the curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)) and their common tangent</a>, 2020.
%H R. A. Rankin, <a href="https://doi.org/10.2307/3608356">2743. An inequality</a>, Mathematical Gazette, 42(339) (1958), 39-40.
%H B. A. Troesch, <a href="https://doi.org/10.1090/S0025-5718-1989-0983563-0">The validity of Shapiro's cyclic inequality</a>, Mathematics of Computation, 53 (1989), 657-664.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ShapirosCyclicSumConstant.html">Shapiro's Cyclic Sum Constant</a>.
%F Equals exp(-c), where c = A319569.
%F Equals the negative of the slope of the common tangent = -(A335339 - A335245)/(A319569 - (-A319568)) = -(exp(-c) - 2/(exp(b) + exp(b/2))) / (c - b).
%e 0.856248214449266116843345...
%o (PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2));
%o a = c(solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2))));
%o exp(-a)
%Y Cf. A086277, A245330, A319568 (negative of x-coordinate at other curve), A319569 (x-coordinate), A335245 (y-coordinate at other curve).
%K nonn,cons
%O 0,1
%A _Petros Hadjicostas_, Jun 02 2020
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