OFFSET
1,2
COMMENTS
rho_3 := +2*cos(5*Pi/7) is the negative zero of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7) = 2*cos(Pi/7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The positive zeros are rho(7) and rho_2 = 2*cos(3*Pi/7) shown in A160389 and A255241.
LINKS
FORMULA
2*cos(5*Pi/7) = - 2*sin(3*Pi/14) = -1.246979603...
Solution of x^3 + x^2 - 2 x - 1 = 0; +1.246979603... - Clark Kimberling, Jan 04 2020
Equals i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 20 2021: (Start)
Equals z + z^6, where z = exp(2*Pi*i/7), so this constant is one of the three cubic Gaussian periods for the modulus 7. The other periods are - A255241 and - A160389.
Equals (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) - 2.
Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = A231187 - 1. (End)
Equals Product_{k>=1} (1 - (-1)^k/A047385(k)). - Amiram Eldar, Nov 22 2024
EXAMPLE
1.2469796037174670610500097680084796212645494617928042107310988781937073049...
MATHEMATICA
r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 210]
RealDigits[r][[1]]
Plot[1/x + 1/(x+1)^2, {x, 1, 2}] (* Clark Kimberling, Jan 04 2020 *)
PROG
(PARI) polrootsreal(x^3 + x^2 - 2*x - 1)[3] \\ Charles R Greathouse IV, Oct 30 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Mar 13 2015
STATUS
approved