A302714 - OEIS

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A302714

Decimal expansion of 2*sin(Pi/192).

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

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OFFSET

-1,1

COMMENTS

This constant appears in a problem similar to the ones posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593. See the Havil reference, pp. 69-74, problem 2. See the comments on A302713 and A302711, also for the Romanus link. The present identity is R(45, 2*sin(Pi/192)) = 2*sin(15*Pi/64) = A302713, with the monic Chebyshev polynomial R from A127672.

This number has been given in Viète's 1595 reply (see A303982 for the link) to Romanus's problems in a corrected Exemplum secundum as solution to the polynomial value given there, which is, in trigonometric version, 2*sin(43*Pi/128) = A303982. Therefore his corrected value (the present one) is also incorrect because it is a solution to the polynomial value 2*sin(15*Pi/64). - Wolfdieter Lang, May 04 2018

REFERENCES

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

LINKS

Table of n, a(n) for n=-1..99.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

Equals sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3)))))).

EXAMPLE

0.03272346325297356328594384696834610047132981567239244974814...

MATHEMATICA

RealDigits[2*Sin[Pi/192], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

CROSSREFS