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%I #33 Feb 11 2021 09:20:19
%S 9,8,9,1,3,3,6,3,4,4,4,6,9,9,3,0,5,2,2,4,3,4,9,0,3,0,8,2,6,6,3,3,7,9,
%T 8,1,3,8,0,3,4,8,0,9,8,0,4,4,1,8,2,2,1,9,0,3,9,3,5,7,8,7,8,0,8,7,3,8,
%U 2,8,9,5,4,2,6,7,5,7,9,5,8,1,5,3,8,0,3,7,6,7,5,0,8,8,0,8,0,3,8,9,4,2,4
%N Decimal expansion of phi(0), an auxiliary constant associated with 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 = 2*A086277. 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 For more information, see my comments in sequence A319568. Based on those comments, we may create the PARI program below that may be used to calculate phi(0). (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 lambda = A086277 = phi(0)/2.
%F phi(0) = exp(-c)*(c+1), where c = A319569. - _Petros Hadjicostas_, Jun 01 2020
%e 0.989133634446993052243490308266337981380348098044182219039357878...
%t digits = 103; phi[0] = (2 + x + 2*E^(x/2)*(1+x))/(E^(x/2)*(1+E^(x/2))^2) /. FindRoot[E^(x/2)*(1+E^(x/2))^2/(1+2*E^(x/2)) == E^(1/(1+2*E^(x/2)) + x), {x, 0}, WorkingPrecision -> digits+10]; RealDigits[phi[0], 10, digits] // First
%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) \\ _Petros Hadjicostas_, Jun 02 2020
%Y Cf. A086277, A086278, A319568, A319569.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jul 18 2014
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