cp's OEIS Frontend

A069855 Decimal expansion of the root of x*tan(x)=1.

%I A069855 #33 Feb 16 2025 08:32:46
%S A069855 8,6,0,3,3,3,5,8,9,0,1,9,3,7,9,7,6,2,4,8,3,8,9,3,4,2,4,1,3,7,6,6,2,3,
%T A069855 3,3,4,1,1,8,8,4,3,6,3,2,3,7,6,5,3,7,8,3,0,0,3,3,8,1,2,8,5,9,0,0,4,0,
%U A069855 3,5,5,0,7,7,2,5,8,0,2,2,1,2,3,3,4,3,0,0,8,5,7,2,1,7,1,4,2,0,8,9,1,7,4,5
%N A069855 Decimal expansion of the root of x*tan(x)=1.
%C A069855 Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and the points A = (u, v), B = (-u, v), C = (-u, -v), D = (u, -v), K = (0, 2v), L = (-2u, 0), M = (0, -2v), N = (2u,0), where u is given by this sequence, and v = u/sqrt(1+u^2). Then ABCD is the rectangle of maximal area, inscribed in S, with sides parallel to the coordinate axes, and KLMN is the rhombus of minimal area, circumscribed around S, with vertices on the coordinate axes. Also, A,B,C,D are the tangent points where the sides of the rhombus touch S, see illustration in the links section. - _Gleb Koloskov_, Jul 05 2021
%H A069855 Gleb Koloskov, <a href="/A346062/a346062.pdf">Geometric illustration</a>
%H A069855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cotangent.html">Cotangent</a> [From _Eric W. Weisstein_, Mar 03 2010]
%F A069855 Equals A346062 * sqrt(2 + 2*sqrt(1 + 256/A346062^2)) / 16. - _Gleb Koloskov_, Jul 05 2021
%e A069855 0.860333589019379762483893424137662333411884363237653783...
%t A069855 N[Minimize[{(x+Cot[x])^2 Sin[x],{x>0,x<Pi/2}},x][[2]],300][[1]][[2]] (* _Gleb Koloskov_, Jul 05 2021 *)
%t A069855 RealDigits[x/.FindRoot[x Tan[x]==1,{x,1},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Dec 04 2021 *)
%o A069855 (PARI) /* 300 significant digits */ s=0.1; for(n=1,500,s=s+sign(cotan(s)-s)/2^n; if(n>499, print(s*1.)))
%Y A069855 Cf. A009399, A085984, A346062.
%K A069855 cons,easy,nonn
%O A069855 0,1
%A A069855 _Benoit Cloitre_, May 01 2002