A255249 Decimal expansion of -2*cos(5*Pi/7).

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

Offset: 1

Author

Wolfdieter Lang, Mar 13 2015

Keywords

• nonn
• cons

Comments

rho_3 := +2*cos(5*Pi/7) is the negative zero of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7) = 2*cos(Pi/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 positive zeros are rho(7) and rho_2 = 2*cos(3*Pi/7) shown in A160389 and A255241.

Essentially the same as A231187 and A116425. - R. J. Mathar, Mar 14 2015

Examples

			1.2469796037174670610500097680084796212645494617928042107310988781937073049...
		

References

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

Links

• Wikipedia, Gaussian period.
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A047385, A160389, A187360, A255241, A330002, A330003 (Beatty sequences), A362922.

Programs

Mathematica

r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 210]
RealDigits[r][[1]]
Plot[1/x + 1/(x+1)^2, {x, 1, 2}] (* Clark Kimberling, Jan 04 2020 *)

PARI

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

Formula

2*cos(5*Pi/7) = - 2*sin(3*Pi/14) = -1.246979603...

Solution of x^3 + x^2 - 2 x - 1 = 0; +1.246979603... - Clark Kimberling, Jan 04 2020

Equals i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020

From Peter Bala, Oct 20 2021: (Start)

Equals z + z^6, where z = exp(2*Pi*i/7), so this constant is one of the three cubic Gaussian periods for the modulus 7. The other periods are - A255241 and - A160389.

Equals (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) - 2.

Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = A231187 - 1. (End)

Equals Product_{k>=1} (1 - (-1)^k/A047385(k)). - Amiram Eldar, Nov 22 2024

Equals 1/(A160389-1) = 2*A362922. - Hugo Pfoertner, Nov 22 2024