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A319568 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 (negated) x-coordinate of the point at which the line touches y = 2/(exp(x) + exp(x/2)). 6
2, 0, 0, 8, 1, 1, 0, 6, 5, 8, 6, 1, 8, 2, 1, 6, 4, 9, 9, 8, 3, 8, 3, 3, 5, 6, 3, 6, 2, 8, 0, 3, 0, 1, 5, 7, 7, 2, 3, 6, 4, 7, 4, 9, 6, 5, 8, 6, 3, 6, 8, 3, 1, 3, 3, 8, 7, 0, 3, 5, 2, 8, 5, 8, 4, 4, 9, 2, 7, 9, 0, 8, 2, 7, 9, 5, 1, 1, 6, 3, 9, 3, 8, 9, 8, 6, 0, 5, 6, 8, 2, 0, 0, 2, 8, 5, 0, 3, 6, 3, 5, 1, 8, 9, 6, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

From Petros Hadjicostas, Jun 01 2020: (Start)

The calculations in this sequence and in A319569 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).

Here phi(x) is the convex hull of y = exp(-x) and y = 2/(exp(x) + exp(x/2)); i.e., phi(x) = 2/(exp(x) + exp(x/2)) for x <= b; phi(x) = exp(-c) + ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) * (x - c) for b <= x <= c; and phi(x) = exp(-x) for x >= c. (For b <= x <= c, we have the equation of the line segment tangent to both curves.)

As stated below, b = -0.200811... (current sequence) and c = 0.15519... (A319569).

It follows that phi(0) = exp(-c) - c * ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) (where the y-axis crosses the line segment). Or by using the tangent line at x = c to the curve y = exp(-x), we find phi(0) = exp(-c)*(c + 1). Or by using the tangent line at x = b to the curve y = 2/(exp(x) + exp(x/2)), we may get a third formula for phi(0).

By using the above information, we get a system of two equations with two unknowns (b and c). See the PARI program below that may be used to calculate b.

Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278. The corresponding curves are y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), while the corresponding x-coordinates at the tangent points are b' = -0.33060... and c' = 0.38875... (not in the OEIS yet). Here psi(x) is the convex hull of these two curves (and it becomes a line segment tangent to both curves for b' <= x <= c').

Eric W. Weisstein, in the link below, has a summary of the above discussion (with contributions by Steven Finch). (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... .

c =  0.1551949747226... (A319569).

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

EXAMPLE

-0.2008110658618...

PROG

(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2))

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, A319569.

Sequence in context: A160213 A192058 A021502 * A334411 A028698 A013667

Adjacent sequences:  A319565 A319566 A319567 * A319569 A319570 A319571

KEYWORD

nonn,cons

AUTHOR

Seiichi Manyama, Sep 23 2018

EXTENSIONS

More terms from Vaclav Kotesovec, Sep 26 2018

STATUS

approved

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Last modified July 25 15:30 EDT 2021. Contains 346291 sequences. (Running on oeis4.)