A243261 Decimal expansion of a Shapiro-Drinfeld constant, known as Gauchman's constant, related to the difference of cyclic sums (negated).

0, 2, 1, 9, 8, 7, 5, 2, 1, 8, 1, 3, 3, 5, 3, 7, 7, 9, 7, 9, 8, 1, 7, 2, 0, 4, 0, 0, 2, 1, 3, 1, 7, 3, 1, 9, 0, 6, 7, 4, 6, 1, 3, 6, 4, 6, 5, 4, 0, 8, 5, 8, 1, 9, 0, 5, 0, 4, 6, 9, 5, 7, 9, 1, 6, 5, 4, 0, 0, 5, 5, 0, 7, 4, 1, 9, 2, 8, 9, 3, 0, 2, 4, 9, 9, 4, 4, 3, 3, 1, 0, 1, 4, 7, 9, 6, 0, 7, 3, 4

Offset

0,2

Comments

Let a_i > 0 for i = 1, ..., n, and define the cyclic sums E_n = a_1/(a_2 + a_3) + a_2/(a_3 + a_4) + ... + a_n/(a_1 + a_2) and F_n = a_1/(a_1 + a_2) + a_2/(a_2 + a_3) + ... + a_n/(a_n + a_1). Gauchman (1998) proved that E_n - F_n >= lambda*n for all n >= 1, where lambda = -0.02198... (this constant) is the best constant. His proof was not published, however, as a solution to Problem 10528(b) in the American Mathematical Monthly (see the link below). Only a comment was made on p. 474. - Petros Hadjicostas, Jun 02 2020

Named after the mathematicians Harold Seymour Shapiro (1928-2021) and Vladimir Drinfeld (b. 1954). Alternatively, named after the American mathematician Hillel V. Gauchman (1937-2016). - Amiram Eldar, May 29 2021

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210.

Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.

Links

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

Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), American Mathematical Monthly, 105 (1998), 473-474. [A comment was made about Hillel Gauchman's solution to part (b) of the problem that involves this constant, but no solution was published.]

Petros Hadjicostas, Plot of the curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 and their common tangent, 2020.

Formula

From Petros Hadjicostas, Jul 04 2020: (Start)

We solve the following system of equations:

exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and

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

Then the constant equals (exp(-c)*(1 + c) - 1)/2.

It turns out that b = -A335810 = -0.387552... and c = A335809 = 0.330604... even though A335810 and A335809 are also involved in the calculation of the Shapiro cyclic sum constant mu = A086278.

As a result, this constant equals A086278 - 1 (i.e., without the negation, A086278 + A243261 = 1). (End)

Example

-0.02198752181335377979817204... = -1 + 0.9780124781866462202018...

Mathematica

eq = E^(u/2) + 2*E^u + E^(3*u/2) + E^(u + v) == E^v + 2*E^(u/2 + v) && 2*(y + E^(u/2 - v) + 1) == (u - 2)/E^v + 4/E^(u/2) && u + 6*E^(u/2) + 4*E^u + 4*E^(u/2 + v) + 1 == v + 9*E^v; y0 = y /. FindRoot[eq , {{y, 0}, {u, -1/3}, {v, 1/3}}, WorkingPrecision -> 105]; RealDigits[y0, 10, 99] // First

Prog

(PARI) default("realprecision", 200)
c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2);
a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2)));
(exp(-c(a))*(1 + c(a)) - 1)/2 \\ Petros Hadjicostas, Jul 04 2020

Crossrefs

Cf. A086277, A086278, A335809, A335810.

Sequence in context: A188659 A042977 A198106 * A230358 A228721 A108290

Adjacent sequences: A243258 A243259 A243260 * A243262 A243263 A243264

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 02 2014

Extensions

Name edited by Petros Hadjicostas, Jun 02 2020

Status

approved