|
%I #47 Jul 13 2020 10:08:02
%S 1,1,7,2,3,8,4,9,1,9,9,6,2,1,1,9,7,1,6,5,7,2,2,3,8,9,6,0,7,0,4,6,3,2,
%T 0,2,0,2,2,4,8,0,8,9,1,1,8,6,1,1,1,9,7,7,6,8,0,5,3,2,7,5,8,0,2,9,7,7,
%U 2,4,4,0,2,0,6,8,8,1,7,6,8,6,7,8,4,0,9,9,2,9,5,3,1,2,5,2,5,7,8,1,2,8,4,1,4,7,6
%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 = (1 - exp(x/2))/(exp(x) + exp(x/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%D Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.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 (1 - exp(b/2))/(exp(b) + exp(b/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 also equals A335825 - 1.
%e 0.11723849199621197165722389607046320202248089118...
%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 (1 - exp(a/2))/(exp(a) + exp(a/2))
%Y Cf. A086278, A243261, A335809 (c), A335810 (-b), A335825 (1 plus the constant), A336029 (y-coordinate for c).
%K nonn,cons
%O 0,3
%A _Petros Hadjicostas_, Jul 05 2020
|
