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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)).
7
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
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.
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
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jun 25 2020
STATUS
approved