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 A231187 #49 Feb 04 2025 10:19:28
%S A231187 2,2,4,6,9,7,9,6,0,3,7,1,7,4,6,7,0,6,1,0,5,0,0,0,9,7,6,8,0,0,8,4,7,9,
%T A231187 6,2,1,2,6,4,5,4,9,4,6,1,7,9,2,8,0,4,2,1,0,7,3,1,0,9,8,8,7,8,1,9,3,7,
%U A231187 0,7,3,0,4,9,1,2,9,7,4,5,6,9,1,5,1,8,8,5,0,1,4,6,5,3,1,7,0,7,4,3,3,3,4,1
%N A231187 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).
%C A231187 The length ratio (largest diagonal)/side in the regular 7-gon (heptagon) is sigma(7) = S(2, rho(7)) = -1 + rho(7)^2, with rho(7) = 2*cos(Pi/7), which is approx. 1.8019377358 (see A160389 for its decimal expansion, and A049310 for the Chebyshev S-polynomials). sigma(7), approx. 2.2469796, is also the reciprocal of one of the solutions of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of rho(7) (see A187360), namely 1/(2*cos(3*Pi/7)).
%C A231187 sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon.
%C A231187 See the Steinbach link.
%H A231187 Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden Fields: A Case for the Heptagon</a>, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%H A231187 I. J. Zucker, G. S. Joyce, <a href="https://doi.org/10.1017/S0305004101005254">Special values of the hypergeometric series II</a>, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq. (8.10).
%H A231187 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A231187 sigma(7) = -1 + (2*cos(Pi/7))^2 = 1/(2*cos(3*Pi/7)).
%F A231187 Equals A116425 -1.
%F A231187 From _Geoffrey Caveney_, Apr 23 2014: (Start)
%F A231187 sigma(7) = exp(asinh(cos(Pi/7))).
%F A231187 cos(Pi/7) + sqrt(1+cos(Pi/7)^2). (End)
%F A231187 From _Peter Bala_, Oct 12 2021: (Start)
%F A231187 Minimal polynomial x^3 - 2*x^2 - x + 1.
%F A231187 Equals 2*(cos(3*Pi/7) - cos(6*Pi/7)). The other zeros of the minimal polynomial are 2*(cos(Pi/7) - cos(2*Pi/7)) = A255240 and 2*(cos(5*Pi/7) - cos(10*Pi/7)) = 1 - A160389.
%F A231187 The quadratic mapping z -> z^2 - 2*z cyclically permutes the zeros of the minimal polynomial. The inverse cyclic permutation is given by the mapping z -> 2 + z - z^2.
%F A231187 Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+1)*(7*n+6)) = 1 + Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = 1 + A255249 = 1/A255241. (End)
%F A231187 Equals 1/(2*sin(Pi/14)) = 1 + 2*sin(3*Pi/14). - _Gary W. Adamson_, Jun 25 2022
%F A231187 Equals (2*cos(Pi/7)) * (2*cos(2*Pi/7)) = (i^(2/7) + i^(-2/7)) * (i^(4/7) + i^(-4/7)) = 1 + i^(4/7) + i^(-4/7). - _Gary W. Adamson_, Jul 16 2022
%F A231187 Equals 2F1(1/7,2/7;1/2;1) [Zucker] - _R. J. Mathar_, Jun 24 2024
%e A231187 2.24697960371746706105000976800847962126454946179280421073109887819...
%t A231187 First[RealDigits[N[Csc[Pi/14]/2,104]]] (* _Stefano Spezia_, Jun 26 2022 *)
%o A231187 (PARI) .5/cos(3*Pi/7) \\ _Charles R Greathouse IV_, Feb 04 2025
%o A231187 (PARI) polrootsreal(x^3-2*x^2-x+1)[3] \\ _Charles R Greathouse IV_, Feb 04 2025
%Y A231187 Cf. A160389, A006054, A077998, A255240, A255241, A255249, A116425.
%K A231187 nonn,cons,easy
%O A231187 1,1
%A A231187 _Wolfdieter Lang_, Nov 21 2013