A319569 - OEIS

%I #55 Feb 11 2021 09:12:50

%S 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,

%T 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,

%U 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

%N 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).

%C From _Petros Hadjicostas_, Jun 01 2020: (Start)

%C 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).

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

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

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld Constant, p. 209.

%H V. G. Drinfel'd, <a href="https://doi.org/10.1007/BF01316982">A cyclic inequality</a>, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.

%H Á. Elbert, <a href="https://doi.org/10.1007/BF02276104">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.

%H Á. Elbert, <a href="https://akjournals.com/view/journals/10998/4/2-3/article-p163.xml">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.

%H Petros Hadjicostas, <a href="/A086277/a086277.pdf">Plot of the curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)) and their common tangent</a>, 2020.

%H R. A. Rankin, <a href="https://doi.org/10.2307/3608356">2743. An inequality</a>, Mathematical Gazette, 42(339) (1958), 39-40.

%H H. S. Shapiro, <a href="https://www.jstor.org/stable/2307617">Proposed problem for solution 4603</a>, American Mathematical Monthly, 61(8) (1954), 571.

%H H. S. Shapiro, <a href="https://www.jstor.org/stable/2306671">Solution to Problem 4603: An invalid inequality</a>, American Mathematical Monthly, 63(3) (1956), 191-192; counterexample provided by M. J. Lighthill.

%H B. A. Troesch, <a href="https://doi.org/10.1090/S0025-5718-1989-0983563-0">The validity of Shapiro's cyclic inequality</a>, Mathematics of Computation, 53 (1989), 657-664.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ShapirosCyclicSumConstant.html">Shapiro's Cyclic Sum Constant</a>.

%F b = -0.2008110658618... (A319568).

%F c =  0.1551949747226... .

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

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

%F Equals -1 - LambertW(-1, -exp(-1)*A245330). - _Vaclav Kotesovec_, Sep 26 2018

%e 0.1551949747226...

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

%o c(solve(b=-2,2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2)))) \\ _Petros Hadjicostas_, Jun 02 2020

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

%K nonn,cons

%O 0,2

%A _Seiichi Manyama_, Sep 23 2018

%E More terms from _Vaclav Kotesovec_, Sep 26 2018