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%I #22 Aug 21 2023 11:51:37
%S 3,6,7,4,9,9,0,3,5,6,3,3,1,4,0,6,6,3,1,4,8,8,1,7,6,7,9,2,4,1,4,5,5,1,
%T 6,4,9,7,8,2,7,7,0,4,7,6,8,8,9,9,8,8,1,1,7,0,1,3,0,1,7,1,5,4,9,7,8,2,
%U 9,8,5,6,5,0,6,1,0,0,3,4,6,0,6,1,2,0,2,3,9,0,2,4,2,1,4,6,0,9,7,1,8,5,9,3,5,9,5
%N Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.
%C s(17) := 2*sin(Pi/17) is an algebraic integer of degree 16 (over the rationals). Its minimal polynomial 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. Its coefficients in the power basis of the algebraic number field Q(2*cos(Pi/34)) are [0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1] (see row l = 8 of A228785). The decimal expansion of 2*cos(Pi/34) is given in A228788.
%C The continued fraction expansion starts with 0; 2, 1, 2, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 2, 1, 1, 43, 3, 1, 5, 2, 17, 2, ...
%C Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into s(17) = sqrt(2*(1 - cos(2*Pi/17))).
%C Since 17 is a Fermat prime, this number is constructible and can be written as an expression containing just integers, the basic four arithmetic operations, and square roots. See A003401 for more details. - _Stanislav Sykora_, May 02 2016
%H Kival Ngaokrajang, <a href="/A228787/a228787.pdf">Illustration of the ratio side/R in the regular 17-gon inscribed in a circle of radius R</a>
%H <a href="/index/Al#algebraic_16">Index entries for algebraic numbers, degree 16</a>
%F s(17) = 2*sin(Pi/17) = 0.367499035633140663148817679...
%F Equals sqrt(34-2*sqrt(17)-2*sqrt(34-2*sqrt(17))-4*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/4. - _Stanislav Sykora_, May 02 2016
%t RealDigits[2Sin[Pi/17], 10, 100][[1]] (* _Alonso del Arte_, Jan 01 2014 *)
%o (PARI) 2*sin(Pi/17) \\ _Charles R Greathouse IV_, Nov 12 2014
%Y Cf. A003401, A019434, A210644, A228785, A228788.
%K nonn,cons
%O 0,1
%A _Wolfdieter Lang_, Oct 07 2013
%E Offset corrected by _Rick L. Shepherd_, Jan 01 2014