A335809 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 (negated) x-coordinate of the point at which the line touches y = (1 + exp(x))/2.

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

Offset

0,1

Comments

It seems that this constant was first calculated by Elbert (1973) in the process of calculating the Shapiro cyclic sum constant mu = A086278 (= the point at which the y-axis intersects the common tangent to the two curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2))). See the discussion in A086278.

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 209.

Links

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

V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.

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 (negated).

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

Also, b = log(2*A335822 - 1).

Example

-0.3306049488870364939879867437698001037271...

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)));
b(a)

Crossrefs

Cf. A086277 (constant lambda), A086278 (constant mu), A243261 (Gauchman's constant), A245330 (2*lambda), A335810 (c), A335822 (y-coordinate for b), A335825 (y-coordinate for c).

Sequence in context: A021972 A193451 A378707 * A322215 A244492 A210838

Adjacent sequences: A335806 A335807 A335808 * A335810 A335811 A335812

Keyword

nonn,cons

Author

Petros Hadjicostas, Jun 24 2020

Status

approved