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A231186 Decimal expansion of the length ratio (largest diagonal)/side in the regular 11-gon (hendecagon). 2
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, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 length ratio (smallest diagonal)/side in the 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 analogon 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.

LINKS

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

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.

CROSSREFS

Sequence in context: A230407 A137759 A049246 * A021078 A130793 A243854

Adjacent sequences:  A231183 A231184 A231185 * A231187 A231188 A231189

KEYWORD

nonn,cons

AUTHOR

Wolfdieter Lang, Nov 20 2013

STATUS

approved

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Last modified March 23 16:52 EDT 2019. Contains 321432 sequences. (Running on oeis4.)