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%I #30 Jun 25 2023 01:56:53
%S 1,7,4,0,1,7,3,9,8,2,2,1,7,4,2,2,8,3,7,3,0,4,5,8,4,8,0,8,9,6,7,6,9,7,
%T 6,8,7,8,2,1,6,5,5,5,7,9,0,5,9,6,5,0,9,7,4,2,1,8,7,8,7,6,4,5,7,0,7,8,
%U 7,2,3,6,2,5,6,3,6,8,3,2,5,7,5,5,8,6,9,2,2,0,8,5,1,5,0,4,9,8,8,7,2
%N Decimal expansion of 2*sin(43*Pi/128).
%C This constant expressed in square roots appears in a historic problem (exemplum secundum) posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593. It is given as the value of the degree 45 polynomial R(45, x) (see A127672 for the R coefficients), a monic Chebyshev polynomial of the first kind. A known 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). In trigonometric version this is R_{2*k+1}(2*sin(theta)) = 2*(-1)^k*sin((2*k+1)*theta). See also the Havil reference, bottom of p. 73 (with P for R, and a missing (-1)^k). For k = 22 this allows us to give all 45 possible x values. One of them is the obvious x1 = 2*sin(43*Pi/(45*128)). Another simpler one is x = 2*sin((37/384)*Pi) given in A303983. In the exemplum secundum, Romanus gives in the first link the wrong solution sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))))) which is in trigonometric version 2*sin((1/128)*Pi). This is not among the 45 solutions. In the second link there is a correction of the x value (the last number is now sqrt(3), not sqrt(2)). This is, in trigonometric version, 2*sin(Pi/192) = A302714. However, this is also not the correct value for the given value of the polynomial.
%C Note that in the Vieta (1595) link, p. 5 (using R. bin. instead of r bin., R. bin.), this exemplum secundum is rewritten with the same polynomial value and the x value sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3)))))) = A302714. As just explained this is incorrect.
%C The correct polynomial value of R(45, x) for the x given by Romanus in the first link (that is, 2*sin((1/128)*Pi)) is, by the above mentioned identity, 2*sin((45/128)*Pi, given in A303984.
%C For the other three exempla of Romanus see also the Havil reference, problems 1 (A302711, A302712), 3 (A179260, A302715), and 4 (A272534, A302716). Problem 2 (A302713, A302714) is another one of this type, but not the one Romanus gave as exemplum secundum.
%H Adriano Romano Lovaniensi, <a href="https://babel.hathitrust.org/cgi/pt?id=ucm.5320258006;view=1up;seq=14 ">Ideae Mathematicae</a>, 1593.
%H Adriano Romano Lovaniensi, <a href="https://archive.org/stream/bub_gb_qinevzxnHFoC#page/n15/mode/2up">Ideae Mathematicae</a>, 1593 [alternative link].
%H Franciscus Vieta, <a href="http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10314282_00001.html">Ad problema quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus Francisci Vietae responsum</a>, Paris, 1595.
%H Walter Van Assche, <a href="https://arxiv.org/abs/2203.10955">Chebyshev polynomials in the 16th century</a>, arXiv:2203.10955 [math.HO], 2022.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F This constant is 2*sin(43*Pi/128) = sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2)))))).
%e 2*sin(43*Pi/128) = 1.740173982217422837304584808967697687821655579059650...
%t RealDigits[2*Sin[43*Pi/128], 10, 120][[1]] (* _Amiram Eldar_, Jun 25 2023 *)
%o (PARI) 2*sin(43*Pi/128) \\ _Altug Alkan_, May 05 2018
%Y Cf. A049310, A127672, A179260, A272534, A302711, A302712, A302713, A302714, A302715, A302716, A303982, A303983, A303984.
%K nonn,cons,easy
%O 1,2
%A _Wolfdieter Lang_, May 04 2018
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