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
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
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jun 25 2020
STATUS
approved