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A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

%I A343055 #38 Aug 29 2025 14:15:44
%S A343055 0,9,8,0,1,7,1,4,0,3,2,9,5,6,0,6,0,1,9,9,4,1,9,5,5,6,3,8,8,8,6,4,1,8,
%T A343055 4,5,8,6,1,1,3,6,6,7,3,1,6,7,5,0,0,5,6,7,2,5,7,2,6,4,9,7,9,8,0,9,3,8,
%U A343055 7,3,0,2,7,8,9,0,8,7,5,3,6,8,0,7,1,1,1,0,7,7,1,4,6,3,1,8,5,5,9,5,5,4,0,7,4,2,0,6,5,2,6,4,4,4,1
%N A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).
%C A343055 An algebraic number of degree 16 and denominator 2. - _Charles R Greathouse IV_, Jan 09 2022
%H A343055 G. C. Greubel, <a href="/A343055/b343055.txt">Table of n, a(n) for n = 0..10000</a>
%H A343055 Michael Penn, <a href="https://www.youtube.com/watch?v=y-8XbDLwzuo">The exact value of sin 5⅝°??</a>, YouTube video, 2021.
%H A343055 Wikipedia, <a href="https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals">Trigonometric constants expressed in real radicals</a>
%H A343055 <a href="/index/Al#algebraic_16">Index entries for algebraic numbers, degree 16</a>
%F A343055 Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).
%F A343055 One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - _R. J. Mathar_, Aug 29 2025
%e A343055 0.09801714032956060199419...
%t A343055 RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* _Amiram Eldar_, Apr 27 2021 *)
%o A343055 (PARI) imag(I^(1/16))
%o A343055 (PARI) sin(Pi/32)
%o A343055 (PARI) sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
%o A343055 (Sage) numerical_approx(sin(pi/32), digits=123) # _G. C. Greubel_, Sep 30 2022
%Y A343055 sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).
%K A343055 nonn,cons,easy,changed
%O A343055 0,2
%A A343055 _Seiichi Manyama_, Apr 04 2021