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A255241
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Decimal expansion of 2*cos(3*Pi/7).
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22
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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
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OFFSET
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0,1
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COMMENTS
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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.
Also the edge length of a regular 14-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 A272487). - Stanislav Sykora, May 01 2016
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LINKS
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FORMULA
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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).
Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).
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EXAMPLE
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0.445041867912628808577805128993589518932711137529089910623974031...
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MATHEMATICA
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RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* Robert Price, May 01 2016 *)
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PROG
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(PARI) 2*sin(Pi/14)
(Magma) R:= RealField(120); 2*Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022
(SageMath) numerical_approx(2*cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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