login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Decimal expansion of m, where y=m*x is the line through (0,0) which meets the curve y=cos(3*x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.
3

%I #9 Apr 10 2021 02:02:21

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

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

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

%N Decimal expansion of m, where y=m*x is the line through (0,0) which meets the curve y=cos(3*x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.

%C See the Mathematica program for a graph.

%C xo=0.9350272884749678361451944275323...

%C yo=0.3301955980451199836007253971727...

%C m=0.35314006565912096755666111412785...

%C |OP|=0.99161744799152518925689622748...

%t c = 3;

%t xo = x /. FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]

%t RealDigits[xo] (* A196996 *)

%t m = Sin[c*xo]/xo

%t RealDigits[m] (* A196997 *)

%t yo = m*xo

%t d = Sqrt[xo^2 + yo^2]

%t Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}], ContourPlot[{y == m*x}, {x, 0, 1.5}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

%Y Cf. A196996, A197000, A197002.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 09 2011