A133741 Decimal expansion of offset at which two unit disks overlap by half each's area.

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, 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, 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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Max Chicky Fang, Closed Form for Half-Area Overlap Offset of 2 Unit Disks, arXiv:2403.10529 [math.GM], 2024.

Eric Weisstein's World of Mathematics, Circle-Circle Intersection

Formula

Equals sqrt(1+A003957) - sqrt(1-A003957) = sqrt(2-2*sqrt(1-A003957^2)) = 2*A086751. - Gleb Koloskov, Feb 26 2021

Example

0.8079455065990344186379234801326308858044719291481968445...

Mathematica

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 *)

Prog

(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
(PARI) d=solve(x=0, 1, cos(x)-x); sqrt(2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Feb 27 2021

Crossrefs

Cf. A003957. Equals twice A086751.

Sequence in context: A183001 A262522 A174849 * A066606 A048729 A003131

Adjacent sequences: A133738 A133739 A133740 * A133742 A133743 A133744

Keyword

nonn,cons

Author

Eric W. Weisstein, Sep 22 2007

Status

approved