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A336011
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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 y-coordinate (negated) of the point at which the line touches y = (1 - exp(x/2))/(exp(x) + exp(x/2)).
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1
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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
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0,3
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COMMENTS
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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 y-axis.
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REFERENCES
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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Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.
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LINKS
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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.]
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FORMULA
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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.
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EXAMPLE
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0.11723849199621197165722389607046320202248089118...
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PROG
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(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))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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