A335825 - OEIS

%I #16 Jul 05 2020 01:21:49

%S 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,

%T 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,

%U 7,2,4,4,0,2,0,6,8,8,1,7,6,8

%N Given the two curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + 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 = (1 + exp(x))/(1 + exp(x/2)).

%H Á. Elbert, <a href="https://doi.org/10.1007/BF02276104">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.

%H Á. Elbert, <a href="https://akjournals.com/view/journals/10998/4/2-3/article-p163.xml">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.

%H Petros Hadjicostas, <a href="/A086278/a086278.pdf">Plot of the curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)) and their common tangent</a>, 2020.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ShapirosCyclicSumConstant.html">Shapiro's cyclic sum constant</a>.

%F Solve the following system of equations to find the x-coordinates of the two points where the common tangent touches the two curves:

%F exp(b) = (-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2 and

%F (exp(b)*(c - b + 1) + 1)*(1 + exp(c/2)) = 2*(1 + exp(c)).

%F This sequence gives the decimal expansion of C = (1 + exp(c))/(1 + exp(c/2)) = (1 + exp(A335810))/(1 + exp(A335810/2)).

%e 1.117238491996211971657223896070463202022...

%o (PARI) default("realprecision", 200)

%o b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);

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

%o (1 + exp(a))/(1 + exp(a/2))

%Y Cf. A086277 (constant lambda), A086278 (constant mu), A243261 (Gauchman's constant), A245330 (2*lambda), A335809 (-b), A335810 (c), A335822 (y-coordinate for b).

%K nonn,cons

%O 1,4

%A _Petros Hadjicostas_, Jun 25 2020