Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #57 Feb 24 2023 05:35:36
%S 2,8,4,4,5,8,5,5,0,4,0,9,8,0,1,8,7,8,1,5,9,2,0,1,0,1,8,1,2,6,9,3,1,7,
%T 4,5,3,3,0,0,5,2,8,3,0,7,8,9,4,6,2,6,9,8,0,4,5,8,7,7,5,0,0,3,0,1,1,8,
%U 9,8,9,5,8,4,8,2,9,2,3,9,7,5,3,8,6,9,4,7,2,3,6,0,6,2,2,7,2,2,1,4,6,7,6,4,6,1,7,2,4,4,7
%N The Square Peg in the Round Hole constant.
%C Given a unit circle and a square of equal area, what is the amount of the square peg shavings (or filings) which would allow the peg to be inserted into the circle? It turns out to be not quite two sevenths.
%D Daniel Zwillinger, Editor, CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, Section 4.6.6 Circles, page 334 & figure 4.18, 2003.
%H G. C. Greubel, <a href="/A194940/b194940.txt">Table of n, a(n) for n = 0..10000</a>
%H 1728 Software Systems, <a href="http://www.1728.org/circsect.htm">Circle Sector, Segment, Chord and Arc Calculator</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircularSegment.html">Circular Segment</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_peg_in_a_round_hole">Square peg in a round hole</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Squaring_the_circle">Squaring the circle</a>.
%H Wikipedia, <a href="http://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/2000px-Squaring_the_circle.svg.png">Diagram of the problem</a>.
%F Area = 4*arccos(sqrt(Pi)/2) - sqrt(Pi*(4-Pi)).
%F Area = Pi + sqrt(2*Pi(2 - sqrt(Pi*(4 - Pi)))) - 4*arcsin(sqrt(Pi/4)). - _Robert G. Wilson v_, Mar 19 2014
%e 0.28445855040980187815920101812693174533005283078946269804587750...
%t RealDigits[ 4*ArcCos[ Sqrt[Pi]/2] - Sqrt[ Pi(4 - Pi)], 10, 111][[1]]
%t RealDigits[Pi + Sqrt[ 2Pi(2 - Sqrt[Pi (4 - Pi)])] - 4 ArcSin[ Sqrt[Pi/4]], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 20 2011 *)
%o (PARI) 4*acos(sqrt(Pi)/2) - sqrt(Pi*(4-Pi)) \\ _G. C. Greubel_, Mar 28 2017
%Y Cf. A000796, A019704, A127454.
%K cons,nonn
%O 0,1
%A _William H. Richardson_ and _Robert G. Wilson v_, Sep 05 2011