

A302713


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


4



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
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OFFSET

1,2


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. 6974, 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).


REFERENCES

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 6974.


LINKS

Table of n, a(n) for n=1..102.
Index entries for sequences related to Chebyshev polynomials.


FORMULA

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


EXAMPLE

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


CROSSREFS

Cf. A049310, A127672, A302711, A302714.
Sequence in context: A306234 A290057 A249790 * A136206 A262979 A011190
Adjacent sequences: A302710 A302711 A302712 * A302714 A302715 A302716


KEYWORD

nonn,cons,easy


AUTHOR

Wolfdieter Lang, Apr 28 2018


STATUS

approved



