A335810 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 x-coordinate of the point at which the line touches y = (1 + exp(x))/(1 + exp(x/2)).

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

Offset

0,1

Comments

This constant was first calculated by Elbert (1973) as 0.388752... (slightly incorrectly) 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 for the sequence A086278 for more details.

P. H. Diananda, the reviewer of Árpád Elbert's article in the Mathematical Reviews (cf. MR0349931), states that Elbert's value of this constant was a misprint, and the correct value is 0.387552...

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..89.

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 c.

mu = A086278 = (2 + 2*(1 - c)*exp(c) + (2 + c)*exp(c/2) + (2 - c)*exp(3*c/2))/(2*(1 + exp(c/2))^2) and b = A335809.

Also, c = 2*log((C + sqrt(C^2 + 4*C - 4))/2), where C = A335825.

Example

0.3875522741406091635132606939529949590972136...

Prog

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

Crossrefs

Cf. A086277 (constant lambda), A086278 (constant mu), A243261 (Gauchman's constant), A245330 (2*lambda), A335809 (-b), A335822 (y-coordinate for b), A335825 (y-coordinate for c).

Sequence in context: A357319 A257822 A201293 * A225016 A121992 A201223

Adjacent sequences: A335807 A335808 A335809 * A335811 A335812 A335813

Keyword

nonn,cons

Author

Petros Hadjicostas, Jun 25 2020

Status

approved