A255241 Decimal expansion of 2*cos(3*Pi/7).

4, 4, 5, 0, 4, 1, 8, 6, 7, 9, 1, 2, 6, 2, 8, 8, 0, 8, 5, 7, 7, 8, 0, 5, 1, 2, 8, 9, 9, 3, 5, 8, 9, 5, 1, 8, 9, 3, 2, 7, 1, 1, 1, 3, 7, 5, 2, 9, 0, 8, 9, 9, 1, 0, 6, 2, 3, 9, 7, 4, 0, 3, 1, 7, 9, 4, 8, 4, 2, 4, 6, 4, 0, 5, 7, 0, 9, 4, 6, 3, 8, 1, 4, 9, 1, 6, 2, 1, 0, 5, 2, 1, 6, 1, 4, 5, 9, 1, 2, 6, 9, 7, 4, 9, 4

Offset: 0

Author

Wolfdieter Lang, Mar 13 2015

Keywords

• nonn
• cons

Comments

This is also the decimal expansion of 2*sin(Pi/14).

rho_2 := 2*cos(3*Pi/7) and rho(7) := 2*cos(Pi/7) (see A160389) are the two positive zeros of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The other zero is negative, rho_3 := 2*cos(5*Pi/n). See -A255249.

Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272487). - Stanislav Sykora, May 01 2016

Examples

			0.445041867912628808577805128993589518932711137529089910623974031...
		

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Wikipedia, Constructible number
• Wikipedia, Regular polygon
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A004169, A160389, A187360, A255249, A232736, A255240.

Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - Stanislav Sykora, May 01 2016

Programs

Magma

R:= RealField(120); 2*Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022

Mathematica

RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI

2*sin(Pi/14)

PARI

polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ Charles R Greathouse IV, Oct 30 2023

SageMath

numerical_approx(2*cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022

Formula

2*cos(3*Pi/7) = 2*sin(Pi/14) = 2*A232736 = 1/A231187 = 0.4450418679...

See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14).

Equals i^(6/7) - i^(8/7). - Peter Luschny, Apr 04 2020

From Peter Bala, Oct 11 2021: (Start)

Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).

Equals 1 - A255240. (End)

Extensions

Offset corrected by Stanislav Sykora, May 01 2016