A233700 Decimal expansion of 1/sin(arctan(1/t)) or t/sin(arctan(t)) where t = 2*Pi: hypotenuse for a right triangle of equal area to a disk.

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

Offset

1,1

Comments

"The great mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle." (Quote from Wikipedia link)

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Area of a disk: Triangle method

Formula

Equals sqrt(1+(2*Pi)^2) = sqrt(1 + (A019692)^2) = sqrt(1 + A212002) = 1/sin(A233527) = A019692/sin(A233528) = 1/cos(A233528) = A019692/cos(A233527).

Example

6.362265131567328393691245440586804410699714985138989686582041617045998587331...

Mathematica

RealDigits[(2*Pi)/Sin[ArcTan[2*Pi]], 10, 120][[1]] (* Harvey P. Dale, Jul 12 2014 *)
RealDigits[ Sqrt[1 + 4*Pi^2], 10, 111][[1]] (* Robert G. Wilson v, Mar 12 2015 *)

Prog

(PARI) sqrt(1+(2*Pi)^2)
(Magma) C<i> := ComplexField(); Sqrt(1 + 4*Pi(C)^2) // G. C. Greubel, Jan 08 2018
(Magma) R:=RealField(110); SetDefaultRealField(R); n:=Sqrt(1+4*Pi(R)^2); Reverse(Intseq(Floor(10^108*n))); // Bruno Berselli, Mar 13 2018
(Julia)
using Nemo
RR = RealField(310)
t = const_pi(RR) + const_pi(RR)
t/sin(atan(t)) |> println # Peter Luschny, Mar 13 2018

Crossrefs

Cf. A233527, A019692, A233528.

Sequence in context: A143506 A248580 A008567 * A195436 A194625 A165065

Adjacent sequences: A233697 A233698 A233699 * A233701 A233702 A233703

Keyword

nonn,cons,nice

Author

John W. Nicholson, Dec 16 2013

Status

approved