A335822 Given the two curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate of the point at which the line touches y = (1 + exp(x))/2.

8, 5, 9, 2, 4, 4, 4, 7, 6, 4, 2, 1, 3, 9, 9, 3, 1, 3, 2, 9, 3, 7, 3, 0, 4, 7, 1, 2, 9, 6, 2, 3, 2, 3, 1, 0, 8, 1, 7, 0, 1, 2, 3, 3, 4, 1, 7, 8, 9, 9, 5, 7, 0, 3, 1, 7, 2, 6, 4, 2, 7, 8, 8, 0, 8, 6, 7, 6, 6, 5, 6, 2, 6, 7, 4, 4, 8, 0, 8, 7, 8, 2, 4, 2, 9, 7, 8, 6, 0, 1, 7, 4, 9, 1, 1, 0, 1

Offset

0,1

Links

Table of n, a(n) for n=0..96.

Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.

Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.

Petros Hadjicostas, Plot of the curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)) and their common tangent, 2020.

Eric Weisstein's World of Mathematics, Shapiro's cyclic sum constant.

Formula

Solve the following system of equations to find the x-coordinates of the two points where the common tangent touches the two curves:

exp(b) = (-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2 and

(exp(b)*(c - b + 1) + 1)*(1 + exp(c/2)) = 2*(1 + exp(c)).

This sequence gives the decimal expansion of B = (1 + exp(b))/2 = (1 + exp(A335809))/2.

mu = A086278 = (1 + exp(b)*(1 - b))/2 = B - (B - 1/2)*log(2*B - 1) and A335810 = c.

The constant B - 1/2 = exp(b)/2 = 0.35924447642... gives the slope of the common tangent.

Example

0.8592444764213993132937304712...

Prog

(PARI) default("realprecision", 200)
b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);
a = solve(c=-1, 1, (exp(b(c))*(c - b(c) + 1) + 1)*(1 + exp(c/2)) - 2*(1 + exp(c)));
(1 + exp(b(a)))/2

Crossrefs

Cf. A086277 (constant lambda), A086278 (constant mu), A243261 (Gauchman's constant), A245330 (2*lambda), A335809 (-b), A335810 (c), A335825 (y-coordinate for c).

Sequence in context: A074071 A100126 A330111 * A233033 A244810 A273985

Adjacent sequences: A335819 A335820 A335821 * A335823 A335824 A335825

Keyword

nonn,cons

Author

Petros Hadjicostas, Jun 25 2020

Status

approved