OFFSET
1,1
COMMENTS
This algebraic number rho(9)^2 of degree 3 is a root of its minimal polynomial x^3 - 6*x^2 + 9*x - 1.
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).
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.
This number rho(9)^2 appears as limit of the quotient of consecutive numbers af various sequences, e.g., A094256 and A094829.
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
FORMULA
Equals (2*cos(Pi/9))^2 = rho(9)^2 = A332437^2.
Equals 2 + i^(4/9) - i^(14/9). - Peter Luschny, Apr 04 2020
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
Constant c = 2 + 2*cos(2*Pi/9). The linear fractional transformation z -> c - c/z has order 9, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z))))))))). - Peter Bala, May 09 2024
EXAMPLE
3.5320888862379560704047853011108333478716649...
MATHEMATICA
RealDigits[(2*Cos[Pi/9])^2, 10, 100][[1]] (* Amiram Eldar, Mar 31 2020 *)
PROG
(PARI) (2*cos(Pi/9))^2 \\ Michel Marcus, Sep 23 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Mar 31 2020
STATUS
approved