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A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

%I A232736 #27 Aug 31 2025 10:16:19
%S A232736 2,2,2,5,2,0,9,3,3,9,5,6,3,1,4,4,0,4,2,8,8,9,0,2,5,6,4,4,9,6,7,9,4,7,
%T A232736 5,9,4,6,6,3,5,5,5,6,8,7,6,4,5,4,4,9,5,5,3,1,1,9,8,7,0,1,5,8,9,7,4,2,
%U A232736 1,2,3,2,0,2,8,5,4,7,3,1,9,0,7,4,5,8,1,0,5,2,6,0,8,0,7,2,9,5,6,3,4,8,7,4,7
%N A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).
%C A232736 The corresponding real part is in A232735.
%C A232736 Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - _Vaclav Kotesovec_, Apr 04 2021
%C A232736 The other 2 roots are -A362922 and A073052. - _R. J. Mathar_, Aug 29 2025
%H A232736 Stanislav Sykora, <a href="/A232736/b232736.txt">Table of n, a(n) for n = 0..1000</a>
%H A232736 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A232736 Equals cos(3*Pi/7). - _G. C. Greubel_, Sep 04 2022
%F A232736 Equals 4*A073052^3 -3*A073052. - _R. J. Mathar_, Aug 29 2025
%F A232736 This^2 + A232735^2 = 1. - _R. J. Mathar_, Aug 31 2025
%e A232736 0.222520933956314404288902564496794759466355568764544955311987...
%t A232736 RealDigits[Cos[3*Pi/7], 10, 120][[1]] (* _G. C. Greubel_, Sep 04 2022 *)
%o A232736 (Magma) R:= RealField(120); Cos(3*Pi(R)/7); // _G. C. Greubel_, Sep 04 2022
%o A232736 (SageMath) numerical_approx(cos(3*pi/7), digits=120) # _G. C. Greubel_, Sep 04 2022
%o A232736 (PARI) sin(Pi/14) \\ _Charles R Greathouse IV_, Feb 04 2025
%o A232736 (PARI) polrootsreal(8*x^3-4*x^2-4*x+1)[2] \\ _Charles R Greathouse IV_, Feb 04 2025
%Y A232736 Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).
%Y A232736 See also A323601.
%K A232736 nonn,cons,easy,changed
%O A232736 0,1
%A A232736 _Stanislav Sykora_, Nov 29 2013