|
|
A319569
|
|
Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the x-coordinate of the point at which the line touches y = exp(-x).
|
|
6
|
|
|
1, 5, 5, 1, 9, 4, 9, 7, 4, 7, 2, 2, 6, 0, 1, 9, 8, 1, 1, 0, 3, 7, 1, 7, 4, 2, 9, 5, 6, 2, 8, 3, 9, 3, 8, 3, 6, 6, 0, 6, 0, 4, 3, 2, 4, 9, 0, 9, 6, 6, 7, 7, 4, 0, 4, 2, 8, 9, 3, 8, 2, 4, 8, 0, 9, 6, 0, 7, 3, 2, 7, 3, 6, 7, 5, 0, 2, 0, 3, 9, 1, 3, 6, 6, 6, 2, 7, 2, 7, 4, 2, 4, 9, 3, 9, 9, 8, 1, 2, 2, 9, 2, 0, 8, 8, 1
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
The calculations in this sequence and in A319568 are needed for the estimation of the Shapiro cyclic sum constant lambda = A086277 = phi(0)/2 = A245330/2. This was done in Drinfel'd (1971).
Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278.
See my comments in sequence A319568. The PARI program below is based on those comments and may be used to calculate c. (End)
|
|
REFERENCES
|
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld Constant, p. 209.
|
|
LINKS
|
V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
|
|
FORMULA
|
c = 0.1551949747226... .
1 - b + c = 2*exp(c)/(exp(b) + exp(b/2)).
exp(-c) * (c + 1) = 0.989133634446... (phi(0)).
|
|
EXAMPLE
|
0.1551949747226...
|
|
PROG
|
(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2))
c(solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2)))) \\ Petros Hadjicostas, Jun 02 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|