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 A336011 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)). 1
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This constant is involved in the calculation of Gauchman's constant -A243261 (which equals A086278 - 1). 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. REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210. Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998. LINKS Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), 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.] Petros Hadjicostas, Plot of the curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 and their common tangent, 2020. FORMULA We solve the following system of equations: exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and 2*(1 - exp(b/2)) = (exp(b) + exp(b/2))*(exp(-c)*(1 + c - b) - 1). Then the constant equals (1 - exp(b/2))/(exp(b) + exp(b/2)). 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). As a result, the constant also equals A335825 - 1. EXAMPLE 0.11723849199621197165722389607046320202248089118... PROG (PARI) default("realprecision", 200) c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2); a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2))); (1 - exp(a/2))/(exp(a) + exp(a/2)) CROSSREFS Cf. A086278, A243261, A335809 (c), A335810 (-b), A335825 (1 plus the constant), A336029 (y-coordinate for c). Sequence in context: A065476 A019945 A335825 * A193027 A248288 A248287 Adjacent sequences: A336008 A336009 A336010 * A336012 A336013 A336014 KEYWORD nonn,cons AUTHOR Petros Hadjicostas, Jul 05 2020 STATUS approved

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Last modified February 5 08:15 EST 2023. Contains 360082 sequences. (Running on oeis4.)