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 A255240 #31 Nov 22 2024 06:29:36
%S A255240 5,5,4,9,5,8,1,3,2,0,8,7,3,7,1,1,9,1,4,2,2,1,9,4,8,7,1,0,0,6,4,1,0,4,
%T A255240 8,1,0,6,7,2,8,8,8,6,2,4,7,0,9,1,0,0,8,9,3,7,6,0,2,5,9,6,8,2,0,5,1,5,
%U A255240 7,5,3,5,9,4,2,9,0,5,3,6,1,8,5,0,8,3,7,8,9,4,7,8,3,8,5,4,0
%N A255240 Decimal expansion of 1/(2*cos(Pi/7)).
%C A255240 This is the decimal expansion of t = 1/rho(7) = 2 + rho(7) - rho(7)^2 with rho(7) = 2*cos(Pi/7) the length ratio of the smaller diagonal and the side of a regular heptagon. See A160389 for the decimal expansion of rho(7).
%C A255240 t satisfies the cubic equation t^3 - 2*t^2 - t + 1 = 0.
%C A255240 t = 1/rho(7) is the slope tan(alpha) appearing in Archimedes's neusis construction of the regular heptagon. The corresponding angle alpha is approximately 29,028 degrees. See the link, Figure 1, also for references.
%C A255240 From _Peter Bala_, Oct 16 2021: (Start)
%C A255240 t = sin(Pi/7)/sin(2*Pi/7). The other roots of the cubic equation t^3 - 2*t^2 - t + 1 = 0 are t_1 = 1/(1 - t) = sin(3*Pi/7)/sin(6*Pi/7) = 2.2469796037... and t_2 = 1/(1 - t_1) = - sin(2*Pi/7)/sin(4*Pi/7) = - 0.8019377358.... Compare with A231187 and A160389.
%C A255240 The algebraic number field Q(t) is a totally real cubic field of discriminant 7^2 and class number 1 with a cyclic Galois group over Q of order 3. See Shanks. (End)
%H A255240 Wolfdieter Lang, <a href="/A255240/a255240.pdf">Archimedes's Construction of the Regular Heptagon</a>.
%H A255240 Daniel 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.
%H A255240 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A255240 1/rho(7) = 1/(2*cos(Pi/7)) = 0.55495813208...
%F A255240 From _Peter Bala_, Oct 10 2021: (Start)
%F A255240 t = 2*(cos(Pi/7) - cos(2*Pi/7)); t_1 = 2*(cos(3*Pi/7) - cos(6*Pi/7)); t_2 = 2*(cos(5*Pi/7) - cos(10*Pi/7)).
%F A255240 t = Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+2)*(7*n+5)) = 1 - Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+3)*(7*n+4)) = 1 - A255241. (End)
%F A255240 Equals Product_{k>=1} (1 + (-1)^k/A047385(k)). - _Amiram Eldar_, Nov 22 2024
%e A255240 0.5549581320873711914221948710064104810672888624709100893760259682051575359...
%t A255240 RealDigits[1/(2*Cos[Pi/7]), 10, 100][[1]] (* _Georg Fischer_, Apr 04 2020 *)
%Y A255240 Cf. A047385, A116425, A160389, A231187, A255241.
%K A255240 nonn,cons
%O A255240 0,1
%A A255240 _Wolfdieter Lang_, Mar 12 2015
%E A255240 Name corrected by _Georg Fischer_, Apr 04 2020