A133741 - OEIS

%I #26 Mar 23 2024 20:03:33

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

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

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

%N Decimal expansion of offset at which two unit disks overlap by half each's area.

%H G. C. Greubel, <a href="/A133741/b133741.txt">Table of n, a(n) for n = 0..10000</a>

%H Max Chicky Fang, <a href="https://arxiv.org/abs/2403.10529">Closed Form for Half-Area Overlap Offset of 2 Unit Disks</a>, arXiv:2403.10529 [math.GM], 2024.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Circle-CircleIntersection.html">Circle-Circle Intersection</a>

%F Equals sqrt(1+A003957) - sqrt(1-A003957) = sqrt(2-2*sqrt(1-A003957^2)) = 2*A086751. - _Gleb Koloskov_, Feb 26 2021

%e 0.8079455065990344186379234801326308858044719291481968445...

%t d0 = d /. FindRoot[ 2*ArcCos[d/2] - d/2*Sqrt[4 - d^2] == Pi/2, {d, 1}, WorkingPrecision -> 110]; RealDigits[d0][[1]][[1 ;; 105]] (* _Jean-François Alcover_, Oct 26 2012, after _Eric W. Weisstein_ *)

%o (PARI) default(realprecision, 100); solve(x=0,1, 2*acos(x/2) - (x/2)*sqrt(4-x^2) - Pi/2) \\ _G. C. Greubel_, Nov 16 2018

%o (PARI) d=solve(x=0,1,cos(x)-x);sqrt(2-2*sqrt(1-d^2)) \\ _Gleb Koloskov_, Feb 27 2021

%Y Cf. A003957. Equals twice A086751.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Sep 22 2007