This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A332438 #53 Aug 17 2025 01:25:03
%S A332438 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 A332438 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 A332438 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 A332438 Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.
%C A332438 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 A332438 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 A332438 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 A332438 This number rho(9)^2 appears as limit of the quotient of consecutive numbers af various sequences, e.g., A094256 and A094829.
%C A332438 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
%H A332438 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A332438 Equals (2*cos(Pi/9))^2 = rho(9)^2 = A332437^2.
%F A332438 Equals 2 + i^(4/9) - i^(14/9). - _Peter Luschny_, Apr 04 2020
%F A332438 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
%F A332438 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
%F A332438 From _Amiram Eldar_, Nov 22 2024: (Start)
%F A332438 Equals 3 + sec(Pi/9)/2 = 3 + 1/(2*A019879).
%F A332438 Equals 3 + Product_{k>=3} (1 + (-1)^k/A063289(k)). (End)
%e A332438 3.5320888862379560704047853011108333478716649...
%t A332438 RealDigits[(2*Cos[Pi/9])^2, 10, 100][[1]] (* _Amiram Eldar_, Mar 31 2020 *)
%o A332438 (PARI) (2*cos(Pi/9))^2 \\ _Michel Marcus_, Sep 23 2022
%Y A332438 Cf. A019879, A063289, A094256, A094829, A127672, A130880, A187360, A332437.
%Y A332438 2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A296184 (n = 10), A019973 (n = 12).
%K A332438 nonn,cons
%O A332438 1,1
%A A332438 _Wolfdieter Lang_, Mar 31 2020