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A319568
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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 (negated) x-coordinate of the point at which the line touches y = 2/(exp(x) + exp(x/2)).
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6
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2, 0, 0, 8, 1, 1, 0, 6, 5, 8, 6, 1, 8, 2, 1, 6, 4, 9, 9, 8, 3, 8, 3, 3, 5, 6, 3, 6, 2, 8, 0, 3, 0, 1, 5, 7, 7, 2, 3, 6, 4, 7, 4, 9, 6, 5, 8, 6, 3, 6, 8, 3, 1, 3, 3, 8, 7, 0, 3, 5, 2, 8, 5, 8, 4, 4, 9, 2, 7, 9, 0, 8, 2, 7, 9, 5, 1, 1, 6, 3, 9, 3, 8, 9, 8, 6, 0, 5, 6, 8, 2, 0, 0, 2, 8, 5, 0, 3, 6, 3, 5, 1, 8, 9, 6, 7
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OFFSET
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0,1
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COMMENTS
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The calculations in this sequence and in A319569 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).
Here phi(x) is the convex hull of y = exp(-x) and y = 2/(exp(x) + exp(x/2)); i.e., phi(x) = 2/(exp(x) + exp(x/2)) for x <= b; phi(x) = exp(-c) + ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) * (x - c) for b <= x <= c; and phi(x) = exp(-x) for x >= c. (For b <= x <= c, we have the equation of the line segment tangent to both curves.)
As stated below, b = -0.200811... (current sequence) and c = 0.15519... (A319569).
It follows that phi(0) = exp(-c) - c * ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) (where the y-axis crosses the line segment). Or by using the tangent line at x = c to the curve y = exp(-x), we find phi(0) = exp(-c)*(c + 1). Or by using the tangent line at x = b to the curve y = 2/(exp(x) + exp(x/2)), we may get a third formula for phi(0).
By using the above information, we get a system of two equations with two unknowns (b and c). See the PARI program below that may be used to calculate b.
Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278. The corresponding curves are y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), while the corresponding x-coordinates at the tangent points are b' = -0.33060... and c' = 0.38875... (not in the OEIS yet). Here psi(x) is the convex hull of these two curves (and it becomes a line segment tangent to both curves for b' <= x <= c').
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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. 209.
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LINKS
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V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
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FORMULA
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b = -0.2008110658618... .
exp(c) = (1 + exp(b/2))^2/(2 + exp(-b/2)).
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EXAMPLE
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-0.2008110658618...
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PROG
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(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2))
solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2))) \\ Petros Hadjicostas, Jun 02 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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