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

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, 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, 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

Offset

0,1

Comments

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

Links

Table of n, a(n) for n=0..103.

Gleb Koloskov, Geometric illustration

Eric Weisstein's World of Mathematics, Cotangent [From Eric W. Weisstein, Mar 03 2010]

Formula

Equals A346062 * sqrt(2 + 2*sqrt(1 + 256/A346062^2)) / 16. - Gleb Koloskov, Jul 05 2021

Example

0.860333589019379762483893424137662333411884363237653783...

Mathematica

N[Minimize[{(x+Cot[x])^2 Sin[x], {x>0, x<Pi/2}}, x][[2]], 300][[1]][[2]] (* Gleb Koloskov, Jul 05 2021 *)
RealDigits[x/.FindRoot[x Tan[x]==1, {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)

Prog

(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.)))

Crossrefs

Cf. A009399, A085984, A346062.

Sequence in context: A010526 A199473 A153617 * A156551 A074738 A344041

Adjacent sequences: A069852 A069853 A069854 * A069856 A069857 A069858

Keyword

cons,easy,nonn

Author

Benoit Cloitre, May 01 2002

Status

approved