A303984 - OEIS

A303984

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

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, 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, 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

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OFFSET

-1,1

COMMENTS

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).

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.

LINKS

Table of n, a(n) for n=-1..106.

Adriano Romano Lovaniensi, Ideae Mathematicae, 1593.

Adriano Romano Lovaniensi, Ideae Mathematicae, 1593 [alternative link].

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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

EXAMPLE

0.04908245704582457606346905891856585013093223847890295515351354757698589627...

MATHEMATICA

RealDigits[2*Sin[Pi/128], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PROG