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

0,1

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)

Links

Table of n, a(n) for n=0..96.

Wolfdieter Lang, Archimedes's Construction of the Regular Heptagon.

D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152

Index entries for algebraic numbers, degree 3

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)

Example

0.5549581320873711914221948710064104810672888624709100893760259682051575359...

Mathematica

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

Crossrefs

Cf. A116425, A160389, A231187, A255241.

Sequence in context: A011189 A261346 A011409 * A168578 A344235 A019253

Adjacent sequences: A255237 A255238 A255239 * A255241 A255242 A255243

Keyword

nonn,cons

Author

Wolfdieter Lang, Mar 12 2015

Extensions

Name corrected by Georg Fischer, Apr 04 2020

Status

approved