A335825 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)).

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, 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

Offset

1,4

Links

Table of n, a(n) for n=1..82.

Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.

Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.

Petros Hadjicostas, Plot of the curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)) and their common tangent, 2020.

Eric Weisstein's World of Mathematics, Shapiro's cyclic sum constant.

Formula

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

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

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

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

Example

1.117238491996211971657223896070463202022...

Mathematica

RealDigits[(1 + Exp[#])/(1 + Exp[#/2]) & @ c /. FindRoot[{Exp[b] == (-Exp[c/2] + 2*Exp[c] + Exp[3*c/2])/(1 + Exp[c/2])^2, (Exp[b]*(c - b + 1) + 1)*(1 + Exp[c/2]) == 2*(1 + Exp[c])}, {{b, 1}, {c, 1}}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Apr 03 2026 *)

Prog

(PARI) default("realprecision", 200)
b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);
a = solve(c=-1, 1, (exp(b(c))*(c - b(c) + 1) + 1)*(1 + exp(c/2)) - 2*(1 + exp(c)));
(1 + exp(a))/(1 + exp(a/2))

Crossrefs

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

Sequence in context: A065476 A019945 A379533 * A336011 A193027 A248288

Adjacent sequences: A335822 A335823 A335824 * A335826 A335827 A335828

Keyword

nonn,cons

Author

Petros Hadjicostas, Jun 25 2020

Status

approved