%I #37 Sep 26 2022 10:32:49
%S 3,5,3,2,0,8,8,8,8,6,2,3,7,9,5,6,0,7,0,4,0,4,7,8,5,3,0,1,1,1,0,8,3,3,
%T 3,4,7,8,7,1,6,6,4,9,1,4,1,6,0,7,9,0,4,9,1,7,0,8,0,9,0,5,6,9,2,8,4,3,
%U 1,0,7,7,7,7,1,3,7,4,9,4,4,7,0,5,6,4,5,8,5,5,3,3,6,1,0,9,6,9
%N Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.
%C This algebraic number rho(9)^2 of degree 3 is a root of its minimal polynomial x^3 - 6*x^2 + 9*x - 1.
%C The other two roots are x2 = (2*cos(5*Pi/9))^2 = (2*cos(4*Pi/9))^2 = (R(4,rho(9))^2 = 2 - rho(9) = 0.120614758..., and x3 = (2*cos(7*Pi/9))^2 = (2*cos(7*Pi/9))^2 = (R(7,rho(9))^2 = 4 + rho(9) - rho(9)^2 = 2.347296355... = A130880 + 2, with rho(9) = 2*cos(Pi/9) = A332437, the monic Chebyshev polynomials R (see A127672), and the computation is done modulo the minimal polynomial of rho(9) which is x^3 - 3*x - 1 (see A187360).
%C This gives the representation of these roots in the power basis of the simple field extension Q(rho(9)). See the linked W. Lang paper in A187360, sect. 4.
%C This number rho(9)^2 appears as limit of the quotient of consecutive numbers af various sequences, e.g., A094256 and A094829.
%C The algebraic number rho(9)^2 - 2 = 1.532088898... of Q(rho(9)) has minimal polynomial x^3 - 3*x + 1 over Q. The other roots are -rho(9) = -A332437 and 2 + rho(9) - rho(9)^2 = A130880. - _Wolfdieter Lang_, Sep 20 2022
%F Equals (2*cos(Pi/9))^2 = rho(9)^2 = A332437^2.
%F Equals 2 + i^(4/9) - i^(14/9). - _Peter Luschny_, Apr 04 2020
%F Equals 2 + w1^(1/3) + w2^(1/3), where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1. - _Wolfdieter Lang_, Sep 20 2022
%e 3.5320888862379560704047853011108333478716649...
%t RealDigits[(2*Cos[Pi/9])^2, 10, 100][[1]] (* _Amiram Eldar_, Mar 31 2020 *)
%o (PARI) (2*cos(Pi/9))^2 \\ _Michel Marcus_, Sep 23 2022
%Y Cf. A094256, A094829, A127672, A130880, A187360, A332437.
%K nonn,cons
%O 1,1
%A _Wolfdieter Lang_, Mar 31 2020
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