%I #27 Jul 13 2020 10:07:58
%S 1,4,0,7,5,5,5,2,3,5,7,8,6,0,0,6,8,6,7,0,6,2,6,9,5,2,8,7,0,3,7,6,7,6,
%T 8,9,1,8,2,9,8,7,6,6,5,8,2,1,0,0,4,2,9,6,8,2,7,3,5,7,2,1,1,9,1,3,2,3,
%U 3,4,3,7,3,2,5,5,1,9,1,2,1,7,5,7,0,2,1,3,9,8,2,5,0,8,8,9,8,8,5,4,9,2,6,0,8,9,5,3,7,7,5,1
%N Given the two curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2, draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate (negated) of the point at which the line touches y = (exp(-x) - 1)/2.
%C This constant is involved in the calculation of Gauchman's constant -A243261 (which equals A086278 - 1).
%C Gauchman's constant is the point where the common tangent to the two curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 intersects the y-axis.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210.
%D Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.
%H Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, <a href="https://www.jstor.org/stable/3109827">Solution to Problem 10528(a,b)</a>, American Mathematical Monthly, 105 (1998), 473-474. [A comment was made about Hillel Gauchman's solution to part (b) of the problem that involves this constant, but no solution was published.]
%H Petros Hadjicostas, <a href="/A243261/a243261.pdf">Plot of the curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 and their common tangent</a>, 2020.
%F We solve the following system of equations:
%F exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and
%F 2*(1 - exp(b/2)) = (exp(b) + exp(b/2))*(exp(-c)*(1 + c - b) - 1).
%F Then the constant equals C = (exp(-c) - 1)/2.
%F It turns out that b = -A335810 = -0.387552... and c = A335809 = 0.330604... even though A335810 and A335809 are also involved in the calculation of the Shapiro cyclic sum constant mu (A086278).
%F As a result, the constant C equals A335822 - 1.
%F Gauchman's constant = -A243261 = C - (C + 1/2)*log(2*C + 1).
%F Shapiro cyclic sum constant mu = A086278 = 1 + C - (C + 1/2)*log(2*C + 1).
%e -0.14075552357860068670626952870376... = -1 + 0.859244476421399313293730471...
%o (PARI) default("realprecision", 200)
%o c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2);
%o a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2)));
%o (exp(-c(a))-1)/2
%Y Cf. A086278, A243261, A335809 (c), A335810 (-b), A335822 (1 plus constant), A336011 (y-coordinate for b).
%K nonn,cons
%O 0,2
%A _Petros Hadjicostas_, Jul 05 2020