A348722 - OEIS

%I #15 Nov 08 2021 16:32:14

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

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

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

%N Decimal expansion of 4*cos(8*Pi/13)*cos(12*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_2.

%C See A348720 and A348721 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 4*cos(Pi/13)*cos(5*Pi/13).

%F Equals 2*(cos(4*Pi/13) + cos(6*Pi/13)).

%F Equals 2*(cos(Pi/13) + cos(5*Pi/13) - cos(2*Pi/13) - cos(10*Pi/13)) - 1.

%F Equals sin(2*Pi/13)*sin(3*Pi/13)/(sin(Pi/13)*sin(5*Pi/13)).

%F Equals Product_{n >= 0} (13*n+2)*(13*n+3)*(13*n+10)*(13*n+11)/( (13*n+1)*(13*n+5)*(13*n+8)*(13*n+12) ).

%F Equivalently, let z = exp(2*Pi*i/13). Then the constant equals abs( (1 - z^2)*(1 - z^3)/((1 - z)*(1 - z^5)) ).

%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)^(4/13) + (-1)^(6/13) - (-1)^(7/13) - (-1)^(9/13). - _Peter Luschny_, Nov 08 2021

%e 1.3772028539729577117217504931601204136143474233621 ...

%p evalf(4*cos(8*Pi/13)*cos(12*Pi/13), 100);

%t RealDigits[4*Cos[8*Pi/13]*Cos[12*Pi/13], 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)

%Y Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

%K nonn,easy,cons

%O 1,2

%A _Peter Bala_, Oct 31 2021