

A336011


Given the two curves y = (1  exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(x)  1)/2, draw a line tangent to both. This sequence is the decimal expansion of the ycoordinate (negated) of the point at which the line touches y = (1  exp(x/2))/(exp(x) + exp(x/2)).


1



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

0,3


COMMENTS

This constant is involved in the calculation of Gauchman's constant A243261 (which equals A086278  1).
Gauchman's constant is the point where the common tangent to the two curves y = (1  exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(x)  1)/2 intersects the yaxis.


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, ShapiroDrinfeld constant, p. 210.
Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.


LINKS

Table of n, a(n) for n=0..106.
Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), American Mathematical Monthly, 105 (1998), 473474. [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

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 (1  exp(b/2))/(exp(b) + exp(b/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, the constant also equals A335825  1.


EXAMPLE

0.11723849199621197165722389607046320202248089118...


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)));
(1  exp(a/2))/(exp(a) + exp(a/2))


CROSSREFS

Cf. A086278, A243261, A335809 (c), A335810 (b), A335825 (1 plus the constant), A336029 (ycoordinate for c).
Sequence in context: A065476 A019945 A335825 * A193027 A248288 A248287
Adjacent sequences: A336008 A336009 A336010 * A336012 A336013 A336014


KEYWORD

nonn,cons


AUTHOR

Petros Hadjicostas, Jul 05 2020


STATUS

approved



