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A302711 Decimal expansion of 2*sin(15*Pi/32). 9


%S 1,9,9,0,3,6,9,4,5,3,3,4,4,3,9,3,7,7,2,4,8,9,6,7,3,9,0,6,2,1,8,9,5,9,

%T 8,4,3,1,5,0,9,4,9,7,3,7,4,5,9,7,1,4,1,2,3,6,6,7,2,2,5,9,3,1,5,6,9,7,

%U 8,0,3,3,3,7,8,9,1,7,3,0,7,5,9,4,5,0,5,8,1,6,8,5,3,9,2,9,6,7,8,0

%N Decimal expansion of 2*sin(15*Pi/32).

%C This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète (see the Vieta link) using trigonometry. See the Havil reference, problem 1 (for a correction see below), pp. 69-74, and the Maor reference for Viète's approach, pp. 58-60.

%C The problem involves the monic Chebyshev polynomial of the first kind R(45, x) (R coefficients are given in A127672). The present problem was stated as R(45, x) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))) for x = (1/2)*sqrt(2 - sqrt(2 + sqrt(2 + sqrt(3)))). This is equivalent to R(45, 2*sin(Pi/96)) = 2*sin(15*Pi/32). It is a special case of the well known identity 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). Take k = 22, x = 2*sin(Pi/96), and see the Havil reference, p. 71, for the proof of 2*sin(15*Pi/32) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))). [In the Havil reference on p. 69, the second to last exponent is 43 (not 41), and in the first problem, for the argument x a furthe +sqrt(2... is missing. In the general identity given on p. 71 a sign factor is missing. It should read, with n = 2*k+1: P_{2*k+1}(2*sin(theta)) = 2*(-1)^k*sin((2*k+1)*theta).]

%C For the argument x = sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3))))) = 2*sin(Pi/96) = 0.65438165643552284... see A302712.

%C R(45, x) factorizes into minimal polynomials of 2*cos(Pi/k), named C(k, x), for short, C[k], with coefficients given in A187360 as follows. R(45, x) = C[90]*C[30]*C[18]*C[10]*C[6]*C[2]. See a comment in A127672.

%C All 45 zeros of R(45, x), which are real, are 2*cos((2*k+1)*Pi/90), for k = 0..44. See a comment in A127672.

%C Viète used the iteration, written in terms of R polynomials as R(45, x) = -R(3, -R(3, R(5, x))) (from the semi-group property of Chebyshev T polynomials). See the Maor reference, pp. 58-60. - _Wolfdieter Lang_, May 05 2018

%D Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

%D Eli Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 56-62.

%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, Paris, 1595</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F This constant is 2*sin(15*Pi/32) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))). (for a proof see Havil. p.71).

%e 2*sin(15*Pi/32) = 1.990369453344393772489673906218959843150949737459714123...

%t RealDigits[2*Sin[15 Pi/32],10,120][[1]] (* _Harvey P. Dale_, Oct 22 2019 *)

%Y Cf. A049310, A127672, A179260, A187360, A272534, A302712, A302713, A302714, A302715, A302716.

%K nonn,cons,easy

%O 1,2

%A _Wolfdieter Lang_, Apr 28 2018

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Last modified November 15 08:37 EST 2019. Contains 329144 sequences. (Running on oeis4.)