A335339 Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate of the point at which the line touches y = exp(-x).

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

Offset

0,1

Links

Table of n, a(n) for n=0..91.

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

Petros Hadjicostas, Plot of the curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)) and their common tangent, 2020.

R. A. Rankin, 2743. An inequality, Mathematical Gazette, 42(339) (1958), 39-40.

B. A. Troesch, The validity of Shapiro's cyclic inequality, Mathematics of Computation, 53 (1989), 657-664.

Eric Weisstein's World of Mathematics, Shapiro's Cyclic Sum Constant.

Formula

Equals exp(-c), where c = A319569.

Equals the negative of the slope of the common tangent = -(A335339 - A335245)/(A319569 - (-A319568)) = -(exp(-c) - 2/(exp(b) + exp(b/2))) / (c - b).

Example

0.856248214449266116843345...

Mathematica

RealDigits[Exp[-c /. FindRoot[{c == b + Exp[b/2]/(2*Exp[b] + Exp[b/2]), Exp[-c]*(1 - b + c) - 2/(Exp[b] + Exp[b/2]) == 0}, {{b, 0}, {c, 0}}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, Apr 03 2026 *)

Prog

(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2));
a = c(solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2))));
exp(-a)

Crossrefs

Cf. A086277, A245330, A319568 (negative of x-coordinate at other curve), A319569 (x-coordinate), A335245 (y-coordinate at other curve).

Sequence in context: A214174 A366587 A154433 * A107828 A256155 A190412

Adjacent sequences: A335336 A335337 A335338 * A335340 A335341 A335342

Keyword

nonn,cons

Author

Petros Hadjicostas, Jun 02 2020

Status

approved