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%I #43 Mar 23 2024 08:24:03
%S 1,8,7,9,3,8,5,2,4,1,5,7,1,8,1,6,7,6,8,1,0,8,2,1,8,5,5,4,6,4,9,4,6,2,
%T 9,3,9,8,7,2,4,1,6,2,6,8,5,2,8,9,2,9,2,6,6,1,8,0,5,7,3,3,2,5,5,4,8,4,
%U 4,2,4,2,1,9,9,1,7,7,8,9,1,7,8,9,9,4,9,1,7,7,9,6,7,5,8,9,6,1,3,4,9
%N Decimal expansion of 2*cos(Pi/9).
%C This algebraic number called rho(9) of degree 3 = A055034(9) has minimal polynomial C(9, x) = x^3 - 3*x - 1 (see A187360).
%C rho(9) gives the length ratio diagonal/side of the smallest diagonal in the regular 9-gon.
%C The length ratio diagonal/side of the second smallest and the third smallest (or the largest) diagonal in the regular 9-gon are rho(9)^2 - 1 = A332438 - 1 and rho(9) + 1, respectively. - _Mohammed Yaseen_, Oct 31 2020
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F rho(9) = 2*cos(Pi/9).
%F Equals (-1)^(-1/9)*((-1)^(1/9) - i)*((-1)^(1/9) + i). - _Peter Luschny_, Mar 27 2020
%F Equals 2*A019879. - _Michel Marcus_, Mar 28 2020
%F Equals sqrt(A332438). - _Mohammed Yaseen_, Oct 31 2020
%F From _Peter Bala_, Oct 20 2021: (Start)
%F The zeros of x^3 - 3*x - 1 are r_1 = -2*cos(2*Pi/9), r_2 = -2*cos(4*Pi/9) and r_3 = -2*cos(8*Pi/9) = 2*cos(Pi/9).
%F The polynomial x^3 - 3*x - 1 is irreducible over Q (since it is irreducible mod 2) with discriminant equal to 3^4, a square. It follows that the Galois group of the number field Q(2*cos(Pi/9)) over Q is cyclic of order 3.
%F The mapping r -> 2 - r^2 cyclically permutes the zeros r_1, r_2 and r_3. The inverse cyclic permutation is given by r -> r^2 - r - 2.
%F The first differences r_1 - r_2, r_2 - r_3 and r_3 - r_1 are the zeros of the cyclic cubic polynomial x^3 - 9*x - 9 of discriminant 3^6.
%F First quotient relations:
%F r_1/r_2 = 1 + (r_3 - r_1); r_2/r_3 = 1 + (r_1 - r_2); r_3/r_1 = 1 + (r_2 - r_3);
%F r_2/r_1 = (r_3 - r_2) - 2; r_3/r_2 = (r_1 - r_3) - 2; r_1/r_3 = (r_2 - r_1) - 2;
%F r_1/r_2 + r_2/r_3 + r_3/r_1 = 3; r_2/r_1 + r_3/r_2 + r_1/r_3 = -6.
%F Thus the first quotients r_1/r_2, r_2/r_3 and r_3/r_1 are the zeros of the cyclic cubic polynomial x^3 - 3*x^2 - 6*x - 1 of discriminant 3^6. See A214778.
%F Second quotient relations:
%F (r_1*r_2)/(r_3^2) = 3*r_2 - 6*r_1 - 8, with two other similar relations by cyclically permuting the 3 zeros. The three second quotients are the zeros of the cyclic cubic polynomial x^3 + 24*x^2 + 3*x - 1 of discriminant 3^10.
%F (r_1^2)/(r_2*r_3) = 1 - 3*(r_2 + r_3), with two other similar relations by cyclically permuting the 3 zeros. (End)
%F Equals i^(2/9) + i^(-2/9). - _Gary W. Adamson_, Jun 25 2022
%F Equals Re((4+4*sqrt(3)*i)^(1/3)). - _Gerry Martens_, Mar 19 2024
%e rho(9) = 1.87938524157181676810821855464946293987241626852892926618...
%t RealDigits[2 * Cos[Pi/9], 10, 100][[1]] (* _Amiram Eldar_, Mar 27 2020 *)
%o (PARI) 2*cos(Pi/9) \\ _Michel Marcus_, Mar 28 2020
%Y Cf. A019879, A055034, A187360, A214778, A215885, A332438.
%K nonn,cons
%O 1,2
%A _Wolfdieter Lang_, Mar 27 2020
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