A231187 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).

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, 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, 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

Offset

1,1

Comments

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)).

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.

See the Steinbach link.

Links

Table of n, a(n) for n=1..104.

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.

I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq. (8.10).

Formula

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

Equals A116425 -1.

From Geoffrey Caveney, Apr 23 2014: (Start)

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

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

From Peter Bala, Oct 12 2021: (Start)

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

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.

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.

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)

Equals 1/(2*sin(Pi/14)) = 1 + 2*sin(3*Pi/14). - Gary W. Adamson, Jun 25 2022

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

Equals 2F1(1/7,2/7;1/2;1) [Zucker] - R. J. Mathar, Jun 24 2024

Example

2.24697960371746706105000976800847962126454946179280421073109887819...

Mathematica

First[RealDigits[N[Csc[Pi/14]/2, 104]]] (* Stefano Spezia, Jun 26 2022 *)

Crossrefs

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

Sequence in context: A001010 A357952 A091966 * A055529 A337723 A222735

Adjacent sequences: A231184 A231185 A231186 * A231188 A231189 A231190

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Nov 21 2013

Status

approved