A086277 Decimal expansion of Shapiro's cyclic sum constant lambda.

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, 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, 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

Offset

0,1

Comments

From Petros Hadjicostas, Jun 01 2020: (Start)

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.

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

For more information, see my comments in A319568. (End)

Links

Table of n, a(n) for n=0..99.

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 World of Mathematics, Shapiro's Cyclic Sum Constant.

Formula

lambda = phi(0)/2 = A245330/2 = exp(-c)*(c+1)/2, where c = A319569. - Petros Hadjicostas, Jun 01 2020

Example

0.4945668...

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A086278, A245330, A319568, A319569.

Sequence in context: A238849 A360697 A197582 * A271948 A068950 A021673

Adjacent sequences: A086274 A086275 A086276 * A086278 A086279 A086280

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 14 2003

Extensions

More terms from Jean-François Alcover, May 16 2014

Status

approved