A348721 - OEIS

%I #19 Nov 08 2021 16:32:09

%S 2,7,3,8,9,0,5,5,4,9,6,4,2,1,7,5,9,4,5,3,1,4,8,9,8,4,4,6,2,7,4,9,4,9,

%T 8,9,5,1,8,0,9,3,6,5,2,3,4,3,4,1,7,5,3,5,4,6,5,5,4,5,1,3,9,1,5,8,8,5,

%U 1,6,9,9,3,5,8,5,8,2,0,7,2,8,7,9,8,7,5,7,6,7,8,3,1,5,2,9,7,8,1,2

%N Decimal expansion of 4*cos(4*Pi/13)*cos(6*Pi/13).

%C Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.

%C In the case a = 1, corresponding to the prime p = 13, Shanks' cyclic cubic is x^3 - x^2 - 4*x - 1 of discriminant 13^2. The three real roots of the cubic are r_0 = 4*cos(2*Pi/13)*cos(3*Pi/13) = 2.6510934089..., r_1 = - 4*cos(4*Pi/13)*cos(6*Pi/13) = - 0.2738905549... and r_2 = - 4*cos(8*Pi/13)*cos(12*Pi/13) = - 1.3772028539.... Here we consider the absolute value of the root r_1.

%C See A348720 and A348722 for the other two roots.

%H T. W. Cusick and Lowell Schoenfeld, <a href="https://doi.org/10.1090/S0025-5718-1987-0866105-8">A table of fundamental pairs of units in totally real cubic fields</a>, Math. Comp. 48 (1987), 147-158 (see case 4 in the Table)

%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152

%F Equals 2*(cos(2*Pi/13) - cos(3*Pi/13)).

%F Equals sin(Pi/13)*sin(5*Pi/13)/(sin(4*Pi/13)*sin(6*Pi/13)).

%F Equals Product_{n >= 0} (13*n+1)*(13*n+5)*(13*n+8)*(13*n+12)/( (13*n+4)*(13*n+6)*(13*n+7)*(13*n+9) ).

%F Equivalently, let z = exp(2*Pi*i/13). Then the constant = abs( (1 - z)*(1 - z^5)/ ((1 - z^4)*(1 - z^6)) ).

%F Note: C = {1, 5, 8, 12} is the subgroup of nonzero cubic residues in the finite field Z_13 with cosets 2*C = {2, 3, 10, 11} and 4*C = {4, 6, 7, 9}.

%F Equals (-1)^(2/13) - (-1)^(3/13) + (-1)^(10/13) - (-1)^(11/13). - _Peter Luschny_, Nov 08 2021

%e 0.27389055496421759453148984462749498951809365234341 ...

%p evalf(4*cos(4*Pi/13)*cos(6*Pi/13), 100);

%t RealDigits[4*Cos[4*Pi/13]*Cos[6*Pi/13], 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)

%Y Cf. A160389, A255240, A255241, A255249, A348720 - A348729.

%K nonn,cons,easy

%O 0,1

%A _Peter Bala_, Oct 31 2021