A130880 Decimal expansion of 2*sin(Pi/18).

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

Offset: 0

Author

R. J. Mathar, Jul 26 2007

Keywords

• cons
• nonn

Comments

Also: a bond percolation threshold probability on the triangular lattice.

Also: the edge length of a regular 18-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 A272488). - Stanislav Sykora, May 01 2016

Examples

			0.347296355333860697703433253538629592...
		

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.18.1, p. 373.

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Steven R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat., Vol. 3, Issue 2-4 (1999), pp. 323-335.
• Wikipedia, Constructible number.
• Wikipedia, Percolation threshold.
• Wikipedia, Regular polygon.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A004169, A019819, A019829, A019889, A063289, A133749, A178959, A332437, A332438.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A272491 (n=19). - Stanislav Sykora, May 01 2016

Programs

Mathematica

RealDigits[N[2Sin[Pi/18], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI

2*sin(Pi/18)

Formula

Equals 2*A019819 = A019829/A019889.

Algebraic number with minimal polynomial over Q equal to x^3 - 3*x + 1, a cyclic cubic, having zeros 2*sin(Pi/18) (= 2*cos(4*Pi/9)), 2*sin(5*Pi/18) (= 2*cos(2*Pi/9)) and -2*sin(7*Pi/18) (= -2*cos(Pi/9)). Cf. A332437. - Peter Bala, Oct 23 2021

Equals 2 + rho(9) - rho(9)^2, an element of the extension field Q(rho(9)), with rho(9) = 2*cos(Pi/9) = A332437 with minimal polynomial x^3 - 3*x - 1 over Q. - Wolfdieter Lang, Sep 20 2022

Equals -1 + Product_{k>=3} (1 - (-1)^k/A063289(k)). - Amiram Eldar, Nov 22 2024

Equals A133749/2 = 1 - A178959. - Hugo Pfoertner, Dec 15 2024