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
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).
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
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jun 24 2020
STATUS
approved