A302713 Decimal expansion of 2*sin(15*Pi/64).

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

Offset: 1

Author

Wolfdieter Lang, Apr 28 2018

Keywords

• nonn
• cons
• easy

Comments

This constant appears in a problem similar to a historic one posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593. See the Havil reference, pp. 69-74, problem 2 (not exemplum secundum of Romanus). See the comment on A302711, also for the Romanus link. In the Havil reference, problem 2, a further sqrt(2... is missing.

The present problem is equivalent to R(45, 2*sin(Pi/192)) =

2*sin(15*Pi/64), with the monic Chebyshev polynomial R from A127672, and for 2*sin(Pi/192) = 0.032723463252973563... see A302714. The general 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). Here k = 22, x = 2*sin(Pi/192).

Examples

			2*sin(15*Pi/64) = 1.3431179096940368012507537008548436064575012643995889977...
		

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

• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A049310, A127672, A302711, A302714.

Programs

Mathematica

RealDigits[2*Sin[15 Pi/64],10,120][[1]] (* Harvey P. Dale, Sep 22 2019 *)

Formula

The constant is 2*sin(15*Pi/64) = sqrt(2-sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2))))).

Root of the equation 2 + (-2 + x) * x^2 * (2 + x) * (-2 + x^2)^2 * (2 - 4*x^2 + x^4)^2 * (2 + x^2 * (-4 + x^2) * (-2 + x^2)^2)^2 = 0. - Vaclav Kotesovec, Apr 30 2018 [This is the polynomial R(32, x). See A127672 for all 32 roots. - Wolfdieter Lang, May 03 2018]

The constant also equals 2*cos(17*Pi/64), one of the roots of R(32, x) (the one for or k = 8 given in A127872). - Wolfdieter Lang, May 03 2018