

A303982


Decimal expansion of 2*sin(43*Pi/128).


4



1, 7, 4, 0, 1, 7, 3, 9, 8, 2, 2, 1, 7, 4, 2, 2, 8, 3, 7, 3, 0, 4, 5, 8, 4, 8, 0, 8, 9, 6, 7, 6, 9, 7, 6, 8, 7, 8, 2, 1, 6, 5, 5, 5, 7, 9, 0, 5, 9, 6, 5, 0, 9, 7, 4, 2, 1, 8, 7, 8, 7, 6, 4, 5, 7, 0, 7, 8, 7, 2, 3, 6, 2, 5, 6, 3, 6, 8, 3, 2, 5, 7, 5, 5, 8, 6, 9, 2, 2, 0, 8, 5, 1, 5, 0, 4, 9, 8, 8, 7, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

This constant expressed in square roots appears in a historic problem (exemplum secundum) posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593. It is given as the value of the degree 45 polynomial R(45, x) (see A127672 for the R coefficients), a monic Chebyshev polynomial of the first kind. A known identity is R(2*k+1, x) = x*(1)^k*S(2*k, sqrt(4x^2)), with the Chebyshev S polynomials (see A049310 for the coefficients). In trigonometric version this is R_{2*k+1}(2*sin(theta)) = 2*(1)^k*sin((2*k+1)*theta). See also the Havil reference, bottom of p. 73 (with P for R, and a missing (1)^k). For k = 22 this allows us to give all 45 possible x values. One of them is the obvious x1 = 2*sin(43*Pi/(45*128)). Another simpler one is x = 2*sin((37/384)*Pi) given in A303983. In the exemplum secundum, Romanus gives in the first link the wrong solution sqrt(2  sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))))) which is in trigonometric version 2*sin((1/128)*Pi). This is not among the 45 solutions. In the second link there is a correction of the x value (the last number is now sqrt(3), not sqrt(2)). This is, in trigonometric version, 2*sin(Pi/192) = A302714. However, this is also not the correct value for the given value of the polynomial.
Note that in the Vieta (1595) link, p. 5 (using R. bin. instead of r bin., R. bin.), this exemplum secundum is rewritten with the same polynomial value and the x value sqrt(2  sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3)))))) = A302714. As just explained this is incorrect.
The correct polynomial value of R(45, x) for the x given by Romanus in the first link (that is, 2*sin((1/128)*Pi)) is, by the above mentioned identity, 2*sin((45/128)*Pi, given in A303984.
For the other three exempla of Romanus see also the Havil reference, problems 1 (A302711, A302712), 3 (A179260, A302715), and 4 (A272534, A302716). Problem 2 (A302713, A302714) is another one of this type, but not the one Romanus gave as exemplum secundum.


LINKS

Table of n, a(n) for n=1..101.
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 Francisci Vietae responsum, Paris, 1595.
Index entries for sequences related to Chebyshev polynomials.


FORMULA

This constant is 2*sin(43*Pi/128) = sqrt(2 + sqrt(2  sqrt(2  sqrt(2  sqrt(2  sqrt(2)))))).


EXAMPLE

2*sin(43*Pi/128) = 1.740173982217422837304584808967697687821655579059650...


PROG

(PARI) 2*sin(43*Pi/128) \\ Altug Alkan, May 05 2018


CROSSREFS

Cf. A049310, A127672, A179260, A272534, A302711, A302712, A302713, A302714, A302715, A302716, A303982, A303983, A303984.
Sequence in context: A013324 A110792 A221388 * A175998 A329091 A306398
Adjacent sequences: A303979 A303980 A303981 * A303983 A303984 A303985


KEYWORD

nonn,cons,easy


AUTHOR

Wolfdieter Lang, May 04 2018


STATUS

approved



