login
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
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Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.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.]
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
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jul 05 2020
STATUS
approved