A245330 Decimal expansion of phi(0), an auxiliary constant associated with Shapiro's cyclic sum constant lambda.

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, 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, 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

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 = 2*A086277. This was done in Drinfel'd (1971).

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

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)

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..102.

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

lambda = A086277 = phi(0)/2.

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

Example

0.989133634446993052243490308266337981380348098044182219039357878...

Mathematica

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

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) \\ Petros Hadjicostas, Jun 02 2020

Crossrefs

Cf. A086277, A086278, A319568, A319569.

Sequence in context: A163930 A157371 A343060 * A269222 A201994 A243277

Adjacent sequences: A245327 A245328 A245329 * A245331 A245332 A245333

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 18 2014

Status

approved