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%I #42 Oct 30 2023 01:36:35
%S 4,4,5,0,4,1,8,6,7,9,1,2,6,2,8,8,0,8,5,7,7,8,0,5,1,2,8,9,9,3,5,8,9,5,
%T 1,8,9,3,2,7,1,1,1,3,7,5,2,9,0,8,9,9,1,0,6,2,3,9,7,4,0,3,1,7,9,4,8,4,
%U 2,4,6,4,0,5,7,0,9,4,6,3,8,1,4,9,1,6,2,1,0,5,2,1,6,1,4,5,9,1,2,6,9,7,4,9,4
%N Decimal expansion of 2*cos(3*Pi/7).
%C This is also the decimal expansion of 2*sin(Pi/14).
%C rho_2 := 2*cos(3*Pi/7) and rho(7) := 2*cos(Pi/7) (see A160389) are the two positive zeros of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(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 other zero is negative, rho_3 := 2*cos(5*Pi/n). See -A255249.
%C Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272487). - _Stanislav Sykora_, May 01 2016
%H Stanislav Sykora, <a href="/A255241/b255241.txt">Table of n, a(n) for n = 0..2000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Constructible_number">Constructible number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_polygon">Regular polygon</a>
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F 2*cos(3*Pi/7) = 2*sin(Pi/14) = 0.4450418679...
%F See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14).
%F Equals i^(6/7) - i^(8/7). - _Peter Luschny_, Apr 04 2020
%F From _Peter Bala_, Oct 11 2021: (Start)
%F Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).
%F Equals 1 - A255240. (End)
%e 0.445041867912628808577805128993589518932711137529089910623974031...
%t RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* _Robert Price_, May 01 2016 *)
%o (PARI) 2*sin(Pi/14)
%o (PARI) polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ _Charles R Greathouse IV_, Oct 30 2023
%o (Magma) R:= RealField(120); 2*Cos(3*Pi(R)/7); // _G. C. Greubel_, Sep 04 2022
%o (SageMath) numerical_approx(2*cos(3*pi/7), digits=120) # _G. C. Greubel_, Sep 04 2022
%Y Cf. A004169, A160389, A187360, A255249, A232736, A255240.
%Y Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - _Stanislav Sykora_, May 01 2016
%K nonn,cons
%O 0,1
%A _Wolfdieter Lang_, Mar 13 2015
%E Offset corrected by _Stanislav Sykora_, May 01 2016
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