This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255241 #48 Mar 16 2025 10:43:40 %S A255241 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, %T A255241 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, %U A255241 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 %N A255241 Decimal expansion of 2*cos(3*Pi/7). %C A255241 This is also the decimal expansion of 2*sin(Pi/14). %C A255241 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. %C A255241 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 %D A255241 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207. %H A255241 Stanislav Sykora, <a href="/A255241/b255241.txt">Table of n, a(n) for n = 0..2000</a> %H A255241 Wikipedia, <a href="http://en.wikipedia.org/wiki/Constructible_number">Constructible number</a> %H A255241 Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_polygon">Regular polygon</a> %H A255241 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a> %F A255241 2*cos(3*Pi/7) = 2*sin(Pi/14) = 2*A232736 = 1/A231187 = 0.4450418679... %F A255241 See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14). %F A255241 Equals i^(6/7) - i^(8/7). - _Peter Luschny_, Apr 04 2020 %F A255241 From _Peter Bala_, Oct 11 2021: (Start) %F A255241 Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7). %F A255241 Equals 1 - A255240. (End) %e A255241 0.445041867912628808577805128993589518932711137529089910623974031... %t A255241 RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* _Robert Price_, May 01 2016 *) %o A255241 (PARI) 2*sin(Pi/14) %o A255241 (PARI) polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ _Charles R Greathouse IV_, Oct 30 2023 %o A255241 (Magma) R:= RealField(120); 2*Cos(3*Pi(R)/7); // _G. C. Greubel_, Sep 04 2022 %o A255241 (SageMath) numerical_approx(2*cos(3*pi/7), digits=120) # _G. C. Greubel_, Sep 04 2022 %Y A255241 Cf. A004169, A160389, A187360, A255249, A232736, A255240. %Y A255241 Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - _Stanislav Sykora_, May 01 2016 %K A255241 nonn,cons %O A255241 0,1 %A A255241 _Wolfdieter Lang_, Mar 13 2015 %E A255241 Offset corrected by _Stanislav Sykora_, May 01 2016