%I #19 Nov 12 2014 09:28:51
%S 1,9,9,1,4,6,8,3,5,2,5,9,0,0,6,9,0,4,3,7,4,2,3,8,2,3,5,7,8,1,0,9,6,3,
%T 5,6,7,8,0,5,4,4,9,2,3,5,2,3,2,5,9,8,3,9,6,7,4,3,8,0,6,0,3,2,6,1,7,4,
%U 1,4,3,1,8,8,3,5,7,0,6,8,1,6,0,7,5,0,9,6,8,4,9,4,7,4,0,2,5,9,6,8,3,4,0,9
%N Decimal expansion of the algebraic integer 2*cos(Pi/34) of degree 16 = A055034(34) (over the rationals), the length ratio (smallest diagonal)/side of a regular 34-gon.
%C rho(34):= 2*cos(Pi/34) is used in the algebraic number field Q(rho(34)) of degree 16 (see A187360) in which s(17) = 2*cos(Pi/17) (for its decimal expansion see A228787), the length ratio side/R of a regular 17-gon inscribed in a circle of radius R, is an integer. See A228787 for this expansion.
%C Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into rho(34) = sqrt(2+sqrt(2+2*cos(2*Pi/17))).
%C The minimal polynomial of rho(34) is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16 (row n=34 polynomial of A187360).
%C The continued fraction expansion starts with 1; 1, 116, 4, 1, 2, 1, 20, 2, 2, 1, 7, 10, 2, 2, 1, 3, 6, 1, 4, 4, 15, ...
%H Vincenzo Librandi, <a href="/A228788/b228788.txt">Table of n, a(n) for n = 1..1000</a>
%H Kival Ngaokrajang, <a href="/A228788/a228788_1.pdf">Illustration of the length ratio (smallest diagonal)/side of a regular 34-gon</a>
%F 2*cos(Pi/34) = 1.99146835259006904374238235781096...
%t RealDigits[2 Cos[Pi/34], 10, 111][[1]] (* _Robert G. Wilson v_, Jul 28 2014 *)
%o (PARI) 2*cos(Pi/34) \\ _Charles R Greathouse IV_, Nov 12 2014
%Y Cf. A055034, A187360, A210644, A228787.
%K nonn,cons
%O 1,2
%A _Wolfdieter Lang_, Oct 07 2013
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