A255240 Decimal expansion of 1/(2*cos(Pi/7)).

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

Offset: 0

Author

Wolfdieter Lang, Mar 12 2015

Keywords

• nonn
• cons

Comments

This is the decimal expansion of t = 1/rho(7) = 2 + rho(7) - rho(7)^2 with rho(7) = 2*cos(Pi/7) the length ratio of the smaller diagonal and the side of a regular heptagon. See A160389 for the decimal expansion of rho(7).

t satisfies the cubic equation t^3 - 2*t^2 - t + 1 = 0.

t = 1/rho(7) is the slope tan(alpha) appearing in Archimedes's neusis construction of the regular heptagon. The corresponding angle alpha is approximately 29,028 degrees. See the link, Figure 1, also for references.

From Peter Bala, Oct 16 2021: (Start)

t = sin(Pi/7)/sin(2*Pi/7). The other roots of the cubic equation t^3 - 2*t^2 - t + 1 = 0 are t_1 = 1/(1 - t) = sin(3*Pi/7)/sin(6*Pi/7) = 2.2469796037... and t_2 = 1/(1 - t_1) = - sin(2*Pi/7)/sin(4*Pi/7) = - 0.8019377358.... Compare with A231187 and A160389.

The algebraic number field Q(t) is a totally real cubic field of discriminant 7^2 and class number 1 with a cyclic Galois group over Q of order 3. See Shanks. (End)

Examples

			0.5549581320873711914221948710064104810672888624709100893760259682051575359...
		

Links

• Wolfdieter Lang, Archimedes's Construction of the Regular Heptagon.
• Daniel Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A047385, A116425, A160389, A231187, A255241.

Programs

Mathematica

RealDigits[1/(2*Cos[Pi/7]), 10, 100][[1]] (* Georg Fischer, Apr 04 2020 *)

Formula

1/rho(7) = 1/(2*cos(Pi/7)) = 0.55495813208...

From Peter Bala, Oct 10 2021: (Start)

t = 2*(cos(Pi/7) - cos(2*Pi/7)); t_1 = 2*(cos(3*Pi/7) - cos(6*Pi/7)); t_2 = 2*(cos(5*Pi/7) - cos(10*Pi/7)).

t = Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+2)*(7*n+5)) = 1 - Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+3)*(7*n+4)) = 1 - A255241. (End)

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

Extensions

Name corrected by Georg Fischer, Apr 04 2020