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A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

%I A232738 #31 Aug 31 2025 11:22:22
%S A232738 1,9,5,0,9,0,3,2,2,0,1,6,1,2,8,2,6,7,8,4,8,2,8,4,8,6,8,4,7,7,0,2,2,2,
%T A232738 4,0,9,2,7,6,9,1,6,1,7,7,5,1,9,5,4,8,0,7,7,5,4,5,0,2,0,8,9,4,9,4,7,6,
%U A232738 3,3,1,8,7,8,5,9,2,4,5,8,0,2,2,5,3,2,5,3,0,9,2,3,4,0,9,0,3,8,1,7,3,0,9,9,2
%N A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).
%C A232738 The corresponding real part is in A232737.
%H A232738 Stanislav Sykora, <a href="/A232738/b232738.txt">Table of n, a(n) for n = 0..1000</a>
%H A232738 Wikipedia, <a href="https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals">Trigonometric constants expressed in real radicals</a>
%H A232738 <a href="/index/Al#algebraic_08">Index entries for algebraic numbers, degree 8</a>
%F A232738 Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - _Seiichi Manyama_, Apr 04 2021
%F A232738 This^2 + A232737^2 = 1.
%F A232738 Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.
%e A232738 0.195090322016128267848284868477022240927691617751954807754502...
%t A232738 RealDigits[Sin[Pi/16],10,120][[1]] (* _Harvey P. Dale_, Sep 01 2018 *)
%o A232738 (PARI) imag(I^(1/8)) \\ _Seiichi Manyama_, Apr 04 2021
%o A232738 (PARI) sin(Pi/16) \\ _Seiichi Manyama_, Apr 04 2021
%o A232738 (PARI) sqrt(2-sqrt(2+sqrt(2)))/2 \\ _Seiichi Manyama_, Apr 04 2021
%o A232738 (Magma) R:= RealField(116); Sin(Pi(R)/16); // _G. C. Greubel_, Sep 20 2022
%o A232738 (SageMath) numerical_approx(sin(pi/16), digits=116) # _G. C. Greubel_, Sep 20 2022
%Y A232738 Cf. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).
%K A232738 nonn,cons,easy,changed
%O A232738 0,2
%A A232738 _Stanislav Sykora_, Nov 29 2013