A319569 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 x-coordinate of the point at which the line touches y = exp(-x).

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

Offset

0,2

Comments

From Petros Hadjicostas, Jun 01 2020: (Start)

The calculations in this sequence and in A319568 are needed for the estimation of the Shapiro cyclic sum constant lambda = A086277 = phi(0)/2 = A245330/2. This was done in Drinfel'd (1971).

Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278.

See my comments in sequence A319568. The PARI program below is based on those comments and may be used to calculate c. (End)

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..105.

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 = 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.

H. S. Shapiro, Proposed problem for solution 4603, American Mathematical Monthly, 61(8) (1954), 571.

H. S. Shapiro, Solution to Problem 4603: An invalid inequality, American Mathematical Monthly, 63(3) (1956), 191-192; counterexample provided by M. J. Lighthill.

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

Eric Weisstein's MathWorld, Shapiro's Cyclic Sum Constant.

Formula

b = -0.2008110658618... (A319568).

c = 0.1551949747226... .

1 - b + c = 2*exp(c)/(exp(b) + exp(b/2)).

exp(-c) * (c + 1) = 0.989133634446... (phi(0)).

Equals -1 - LambertW(-1, -exp(-1)*A245330). - Vaclav Kotesovec, Sep 26 2018

Example

0.1551949747226...

Prog

(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2))
c(solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2)))) \\ Petros Hadjicostas, Jun 02 2020

Crossrefs

Cf. A086277, A086278, A245330 (phi(0)), A319568.

Sequence in context: A300710 A254347 A011094 * A204005 A075298 A370262

Adjacent sequences: A319566 A319567 A319568 * A319570 A319571 A319572

Keyword

nonn,cons

Author

Seiichi Manyama, Sep 23 2018

Extensions

More terms from Vaclav Kotesovec, Sep 26 2018

Status

approved