A086751 Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.

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

Offset: 0

Author

Jonathan R. Anderson (neo__jon(AT)hotmail.com), Jul 30 2003

Keywords

• cons
• nonn
• easy

Comments

Decimal expansion of the number sin(u'), where u' is the number in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0). See A332500. - Clark Kimberling, May 05 2020

Examples

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Links

• Robert P. P. McKone, Table of n, a(n) for n = 0..19999

Crossrefs

Cf. A003957.

Programs

Maple

Digits := 240 ; x := 0.4 ; for i from 1 to 8 do f := sin(2.0*x)+2.0*x-Pi/2.0 ; fp := 2*cos(2*x)+2.0 ; x := x-evalf(f/fp) ; printf("%.120f\n",sin(x)) ; od: x := sin(x) ; read("transforms3") ; CONSTTOLIST(x) ; # R. J. Mathar, May 19 2009

Mathematica

digits = 105; Sin[FindRoot[Sin[2*a]/2+a == Pi/4, {a, 1/2}, WorkingPrecision -> digits][[1, 2]]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)

PARI

solve(x=0, 1, x*sqrt(1-x^2) + asin(x) - Pi/4) \\ Michel Marcus, May 05 2020

PARI

my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/2 \\ Gleb Koloskov, Jun 16 2021

Formula

Define k(n+1) as k(n) - (k(n)*sqrt(1-k(n)^2) + arcsin(k(n)) - Pi/4). The sequence is the decimal expansion of lim_{n -> infinity} k(n).

Equals sqrt(2-2*sqrt(1-d^2))/2, where d = A003957 is the Dottie number. - Gleb Koloskov, Jun 16 2021

Extensions

More terms from Jim Nastos, Sep 05 2003

More digits from R. J. Mathar, May 19 2009