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%I #41 Oct 30 2023 01:34:44
%S 1,2,4,6,9,7,9,6,0,3,7,1,7,4,6,7,0,6,1,0,5,0,0,0,9,7,6,8,0,0,8,4,7,9,
%T 6,2,1,2,6,4,5,4,9,4,6,1,7,9,2,8,0,4,2,1,0,7,3,1,0,9,8,8,7,8,1,9,3,7,
%U 0,7,3,0,4,9,1,2,9,7,4,5,6,9,1,5,1,8,8,5,0,1,4,6,5,3,1,7,0
%N Decimal expansion of -2*cos(5*Pi/7).
%C 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.
%C Essentially the same as A231187 and A116425. - _R. J. Mathar_, Mar 14 2015
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_period">Gaussian period</a>.
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F 2*cos(5*Pi/7) = - 2*sin(3*Pi/14) = -1.246979603...
%F Solution of x^3 + x^2 - 2 x - 1 = 0; +1.246979603... - _Clark Kimberling_, Jan 04 2020
%F Equals i^(4/7) - i^(10/7). - _Peter Luschny_, Apr 04 2020
%F From _Peter Bala_, Oct 20 2021: (Start)
%F 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.
%F Equals (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) - 2.
%F Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = A231187 - 1.
%F (End)
%e 1.2469796037174670610500097680084796212645494617928042107310988781937073049...
%t r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 210]
%t RealDigits[r][[1]]
%t Plot[1/x + 1/(x+1)^2, {x, 1, 2}] (* _Clark Kimberling_, Jan 04 2020 *)
%o (PARI) polrootsreal(x^3 + x^2 - 2*x - 1)[3] \\ _Charles R Greathouse IV_, Oct 30 2023
%Y Cf. A160389, A187360, A255241, A330002, A330003 (Beatty sequences).
%K nonn,cons
%O 1,2
%A _Wolfdieter Lang_, Mar 13 2015
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