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%I A302713 #15 Sep 22 2019 12:42:18
%S A302713 1,3,4,3,1,1,7,9,0,9,6,9,4,0,3,6,8,0,1,2,5,0,7,5,3,7,0,0,8,5,4,8,4,3,
%T A302713 6,0,6,4,5,7,5,0,1,2,6,4,3,9,9,5,8,8,9,9,7,7,6,6,4,2,6,1,4,6,2,1,8,9,
%U A302713 2,9,8,2,3,7,5,8,0,0,2,8,3,0,3,3,4,5,8,0,9,8,6,3,5,6,8,0,8,3,2,1,3,2
%N A302713 Decimal expansion of 2*sin(15*Pi/64).
%C A302713 This constant appears in a problem similar to a historic one posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593. See the Havil reference, pp. 69-74, problem 2 (not exemplum secundum of Romanus). See the comment on A302711, also for the Romanus link. In the Havil reference, problem 2, a further sqrt(2... is missing.
%C A302713 The present problem is equivalent to R(45, 2*sin(Pi/192)) =
%C A302713 2*sin(15*Pi/64), with the monic Chebyshev polynomial R from A127672, and for 2*sin(Pi/192) = 0.032723463252973563... see A302714. The general identity is R(2*k + 1, x) = x*(-1)^k*S(2*k, sqrt(4 - x^2)), with the Chebyshev S polynomials (see A049310 for the coefficients). Here k = 22, x = 2*sin(Pi/192).
%D A302713 Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
%H A302713 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A302713 The constant is 2*sin(15*Pi/64) = sqrt(2-sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2))))).
%F A302713 Root of the equation 2 + (-2 + x) * x^2 * (2 + x) * (-2 + x^2)^2 * (2 - 4*x^2 + x^4)^2 * (2 + x^2 * (-4 + x^2) * (-2 + x^2)^2)^2 = 0. - _Vaclav Kotesovec_, Apr 30 2018 [This is the polynomial  R(32, x). See A127672 for all 32 roots. - _Wolfdieter Lang_, May 03 2018]
%F A302713 The constant also equals 2*cos(17*Pi/64), one of the roots of R(32, x) (the one for or k = 8 given in A127872). - _Wolfdieter Lang_, May 03 2018
%e A302713 2*sin(15*Pi/64) = 1.3431179096940368012507537008548436064575012643995889977...
%t A302713 RealDigits[2*Sin[15 Pi/64],10,120][[1]] (* _Harvey P. Dale_, Sep 22 2019 *)
%Y A302713 Cf. A049310, A127672, A302711, A302714.
%K A302713 nonn,cons,easy
%O A302713 1,2
%A A302713 _Wolfdieter Lang_, Apr 28 2018

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