

A302711


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


9



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, 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, 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
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OFFSET

1,2


COMMENTS

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. 6974, and the Maor reference for Viète's approach, pp. 5860.
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(4x^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).]
For the argument x = sqrt(2  sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3))))) = 2*sin(Pi/96) = 0.65438165643552284... see A302712.
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.
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.
Viète used the iteration, written in terms of R polynomials as R(45, x) = R(3, R(3, R(5, x))) (from the semigroup property of Chebyshev T polynomials). See the Maor reference, pp. 5860.  Wolfdieter Lang, May 05 2018


REFERENCES

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 6974.
Eli Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 5662.


LINKS

Table of n, a(n) for n=1..100.
Adriano Romano Lovaniensi, Ideae Mathematicae, 1593.
Adriano Romano Lovaniensi,Ideae Mathematicae, 1593 [alternative link].
Franciscus Vieta, Ad Problema. Quod omnibus Mathematicis totius orbis construendum proposuit Adrianus Romanus, Paris, 1595
Index entries for sequences related to Chebyshev polynomials.


FORMULA

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


EXAMPLE

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


CROSSREFS

Cf. A049310, A127672, A179260, A187360, A272534, A302712, A302713, A302714, A302715, A302716.
Sequence in context: A176537 A019891 A176536 * A021838 A199960 A257176
Adjacent sequences: A302708 A302709 A302710 * A302712 A302713 A302714


KEYWORD

nonn,cons,easy


AUTHOR

Wolfdieter Lang, Apr 28 2018


STATUS

approved



