A231186 Decimal expansion of the ratio (longest diagonal)/side in a regular 11-gon (or hendecagon).

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

Offset: 1

Author

Wolfdieter Lang, Nov 20 2013

Keywords

• nonn
• cons

Comments

omega(11):= S(4, x) = 1 - 3*x^2 + x^4 with x = rho(11) := 2*cos(Pi/11). See A049310 for Chebyshev s-polynomials. rho(11) is the ratio (shortest diagonal)/side in a regular 11-gon. See the Q(2*cos(Pi/n)) link given in A187360. This is the power basis representation of omega(11) in the algebraic number field Q(2*cos(Pi/11)) of degree 5.

omega(11) = 1/(2*cos(Pi*5/11)) = 1/R(5, rho(11)) with the R-polynomial given in A127672. This follows from a computation of the power basis coefficients of the reciprocal of R(5, x) (mod C(11, x)) = 1+2*x-3*x^2-x^3+x^4, where C(11, x) is the minimal polynomial of rho(11) given in A187360. The result for this reciprocal (mod C(11, x)) is 1 - 3*x^2 + x^4 giving the power base coefficients [1,0,-3,0,1] for omega(11).

omega(11) is the analog in the regular 11-gon of the golden section in the regular 5-gon (pentagon), because it is the limit of a(n+1)/a(n) for n -> infinity of sequences like A038342, A069006, A230080 and A230081.

The ratio diagonal/side of the second and third shortest diagonals in a regular 11-gon are respectively x^2 - 1 and x^3 - 2*x, where x = 2*cos(Pi/11). - Mohammed Yaseen, Nov 03 2020

Examples

			3.51333709166613518878217159629798184207459481777014...
		

Crossrefs

Cf. A038342, A049310, A069006, A127672, A187360, A230080, A230081.

Programs

Mathematica

RealDigits[Sin[5*Pi/11]/Sin[Pi/11], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

Formula

omega(11) = 1 - 3*x^2 + x^4 with x = rho(11) := 2*cos(Pi/11) = 1/(2*cos(Pi*5/11)) = 3.5133370916661... See the comments above.

Equals sin(5*Pi/11)/sin(Pi/11). - Mohammed Yaseen, Nov 03 2020