A228787 Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.

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, 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, 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

Offset: 0

Author

Wolfdieter Lang, Oct 07 2013

Keywords

• nonn
• cons
• changed

Comments

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.

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, ...

Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into s(17) = sqrt(2*(1 - cos(2*Pi/17))).

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

Examples

			0.367499035633140663148817679...
		

Links

• Kival Ngaokrajang, Illustration of the ratio side/R in the regular 17-gon inscribed in a circle of radius R
• Index entries for algebraic numbers, degree 16

Crossrefs

Cf. A003401, A019434, A210644, A228785, A228788.

Programs

Mathematica

RealDigits[2Sin[Pi/17], 10, 100][[1]] (* Alonso del Arte, Jan 01 2014 *)

PARI

2*sin(Pi/17) \\ Charles R Greathouse IV, Nov 12 2014

Formula

s(17) = 2*sin(Pi/17) = 2*A241243.

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

Extensions

Offset corrected by Rick L. Shepherd, Jan 01 2014