

A231186


Decimal expansion of the length ratio (largest diagonal)/side in the regular 11gon (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
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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 spolynomials. rho(11) is the length ratio (smallest diagonal)/side in the regular 11gon. 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 Rpolynomial 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*x3*x^2x^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 11gon of the golden section in the regular 5gon (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



