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Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).
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%I #46 Jun 24 2024 11:31:10

%S 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 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 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 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).

%C 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 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 See the Steinbach link.

%H 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 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).

%F sigma(7) = -1 + (2*cos(Pi/7))^2 = 1/(2*cos(3*Pi/7)).

%F Equals A116425 -1.

%F From _Geoffrey Caveney_, Apr 23 2014: (Start)

%F sigma(7) = exp(asinh(cos(Pi/7))).

%F cos(Pi/7) + sqrt(1+cos(Pi/7)^2). (End)

%F From _Peter Bala_, Oct 12 2021: (Start)

%F Minimal polynomial x^3 - 2*x^2 - x + 1.

%F 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 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 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 Equals 1/(2*sin(Pi/14)) = 1 + 2*sin(3*Pi/14). - _Gary W. Adamson_, Jun 25 2022

%F 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 Equals 2F1(1/7,2/7;1/2;1) [Zucker] - _R. J. Mathar_, Jun 24 2024

%e 2.24697960371746706105000976800847962126454946179280421073109887819...

%t First[RealDigits[N[Csc[Pi/14]/2,104]]] (* _Stefano Spezia_, Jun 26 2022 *)

%Y Cf. A160389, A006054, A077998, A255240, A255241, A255249, A116425.

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Nov 21 2013