A303984 - OEIS

%I #23 Mar 11 2024 01:54:15

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

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

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

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

%C This constant appears in a historic problem posed as exemplum secundum by Adriaan van Roomen (see the first Adranus Romanus link) as erroneous argument x for the polynomial R(45, x), with the monic Chebyshev polynomials of the first kind R (A127672), of value given as 2*sin(43*Pi/128) = A303982 in a version with iterated square-roots given below. However, the correct polynomial value for the present constant x is R(45, x) = 2*sin(45*Pi/128) = A303985. See also comments, references and links for the other three problems in A302711 and A303982, especially for the identity R(45, 2*sin(theta)) = 2*sin(45*theta).

%C Note that in the second Romano link the x value in exemplum secundum differs, it is x = sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3))))))) = 2*sin(Pi/192) = A302714. But this value x does also not solve R(45, x) = 2*sin(43*Pi/128), but R(45, x) = 2*sin(15*Pi/64) = A302713.

%H Adriano Romano Lovaniensi, <a href="https://babel.hathitrust.org/cgi/pt?id=ucm.5320258006;view=1up;seq=14 ">Ideae Mathematicae</a>, 1593.

%H Adriano Romano Lovaniensi, <a href="https://archive.org/stream/bub_gb_qinevzxnHFoC#page/n15/mode/2up">Ideae Mathematicae</a>, 1593 [alternative link].

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%F Equals sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))))).

%e 0.04908245704582457606346905891856585013093223847890295515351354757698589627...

%t RealDigits[2*Sin[Pi/128], 10, 120][[1]] (* _Amiram Eldar_, Jun 26 2023 *)

%o (PARI) 2*sin(Pi/128) \\ _Altug Alkan_, May 06 2018

%Y Cf. A302711, A302713, A302714, A303982, A303985.

%K nonn,cons,easy

%O -1,1

%A _Wolfdieter Lang_, May 06 2018