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A255241 Decimal expansion of 2 cos(3 Pi/7). 9
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

- Stanislav Sykora, May 01 2016 (Start) Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not costructible 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 contructible, which is not true in this case (see A272487). (End)

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Wikipedia, Constructible number

Wikipedia, Regular polygon

FORMULA

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

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

EXAMPLE

0.445041867912628808577805128993589518932711137529089910623974031...

MATHEMATICA

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

PROG

(PARI) 2*sin(Pi/14)

CROSSREFS

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

Edge lengths of other nonconstructible n-gons: A271487 (n=7), A272488 (n=9), A272489 (n=11),A130880 (n=18), A272491 (n=19). - Stanislav Sykora, May 01 2016

Sequence in context: A111481 A111763 A159892 * A200694 A021696 A006581

Adjacent sequences:  A255238 A255239 A255240 * A255242 A255243 A255244

KEYWORD

nonn,cons

AUTHOR

Wolfdieter Lang, Mar 13 2015

EXTENSIONS

Offset corrected by Stanislav Sykora, May 01 2016

STATUS

approved

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Last modified April 30 12:33 EDT 2017. Contains 285668 sequences.