%I #38 Jun 23 2020 04:39:47
%S 4,9,4,5,6,6,8,1,7,2,2,3,4,9,6,5,2,6,1,2,1,7,4,5,1,5,4,1,3,3,1,6,8,9,
%T 9,0,6,9,0,1,7,4,0,4,9,0,2,2,0,9,1,1,0,9,5,1,9,6,7,8,9,3,9,0,4,3,6,9,
%U 1,4,4,7,7,1,3,3,7,8,9,7,9,0,7,6,9,0,1,8,8,3,7,5,4,4,0,4,0,1,9,4
%N Decimal expansion of Shapiro's cyclic sum constant lambda.
%C From _Petros Hadjicostas_, Jun 01 2020: (Start)
%C The calculations in sequences A319568 and A319569 are needed for the estimation of the constant phi(0) = 2*lambda = A245330. This was done in Drinfel'd (1971) even though Rankin (1958) was probably the first to study this constant.
%C Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278.
%C For more information, see my comments in A319568. (End)
%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 World of Mathematics, <a href="http://mathworld.wolfram.com/ShapirosCyclicSumConstant.html">Shapiro's Cyclic Sum Constant</a>.
%F lambda = phi(0)/2 = A245330/2 = exp(-c)*(c+1)/2, where c = A319569. - _Petros Hadjicostas_, Jun 01 2020
%e 0.4945668...
%t eq = E^(x/2)*y + y == x/(1 + E^(x/2)) + (x + 2)/E^(x/2) && x + 1/(1 + 2*E^(x/2)) == Log[(4*E^x*Cosh[x/4]^2)/(1 + 2*E^(x/2))]; y0 = y /. FindRoot[eq, {y, 1}, {x, -1}, WorkingPrecision -> 105]; RealDigits[y0/2, 10, 100] // First (* _Jean-François Alcover_, May 16 2014 *)
%o (PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2));
%o a=c(solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2))));
%o exp(-a)*(a+1)/2 \\ _Petros Hadjicostas_, Jun 02 2020
%Y Cf. A086278, A245330, A319568, A319569.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jul 14 2003
%E More terms from _Jean-François Alcover_, May 16 2014