A302715 Decimal expansion of 2*sin(Pi/120).

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

Offset

-1,1

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 Havil reference, problem 3, pp. 69-74. See also the comments in A302711 with a link to Romanus book, Exemplum tertium.

The solution of the problem uses the special case of an identity R(45, 2*sin(Pi/120)) = 2*sin(3*Pi/8) = A179260 = 1.847759065022..., with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672.

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

This constant is 2*sin(Pi/120) = sqrt(2 - sqrt(2 + sqrt(3/16) + sqrt(15/16) + sqrt(5/8 - sqrt(5/64)))) (this is the rewritten x given in the Havil reference on the bottom of page 69).

Example

0.05235389661574630522122337110822532758678205536021727643757...

Mathematica

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

Crossrefs

Cf. A127672, A179260, A302711.

Sequence in context: A262429 A377815 A097078 * A237200 A021195 A019673

Adjacent sequences: A302712 A302713 A302714 * A302716 A302717 A302718

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Apr 29 2018

Status

approved