A335245 - OEIS

A335245

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 y-coordinate of the point at which the line touches y = 2/(exp(x) + exp(x/2)).

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

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OFFSET

1,3

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld Constant, p. 209.

LINKS

Table of n, a(n) for n=1..95.

V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.

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

R. A. Rankin, 2743. An inequality, Mathematical Gazette, 42(339) (1958), 39-40.

B. A. Troesch, The validity of Shapiro's cyclic inequality, Mathematics of Computation, 53 (1989), 657-664.

Eric Weisstein's World of Mathematics, Shapiro's Cyclic Sum Constant.

FORMULA

Equals 2/(exp(b) + exp(b/2)), where b = -A319568.

Slope of common tangent = (A335339 - A335245)/(A319569 - (-A319568)) = (exp(-c) - 2/(exp(b) + exp(b/2))) / (c - b) = -A335339 = -exp(-c).

EXAMPLE

1.1610777510328318186075949...

PROG

(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2));

y(b) = 2/(exp(b) + exp(b/2));

a=solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2)));

y(a)

CROSSREFS

Cf. A086277, A245330, A319568 (negative of x-coordinate), A319569 (x-coordinate at other curve), A335339 (y-coordinate at other curve).

Sequence in context: A074395 A355415 A262704 * A195402 A176402 A204013

Adjacent sequences: A335242 A335243 A335244 * A335246 A335247 A335248

KEYWORD

nonn,cons

AUTHOR

Petros Hadjicostas, Jun 02 2020

STATUS

approved