A336029 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.

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, 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, 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

Offset

0,2

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Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.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

Table of n, a(n) for n=0..111.

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 C = (exp(-c) - 1)/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 C equals A335822 - 1.

Gauchman's constant = -A243261 = C - (C + 1/2)*log(2*C + 1).

Shapiro cyclic sum constant mu = A086278 = 1 + C - (C + 1/2)*log(2*C + 1).

Example

-0.14075552357860068670626952870376... = -1 + 0.859244476421399313293730471...

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)));
(exp(-c(a))-1)/2

Crossrefs

Cf. A086278, A243261, A335809 (c), A335810 (-b), A335822 (1 plus constant), A336011 (y-coordinate for b).

Sequence in context: A057641 A352886 A379041 * A272876 A133930 A077892

Adjacent sequences: A336026 A336027 A336028 * A336030 A336031 A336032

Keyword

nonn,cons

Author

Petros Hadjicostas, Jul 05 2020

Status

approved