A199170 - OEIS

A199170

Decimal expansion of x<0 satisfying x^2+x*cos(x)=1.

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

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OFFSET

1,3

COMMENTS

For many choices of a,b,c, there are exactly two numbers x satisfying a*x^2+b*x*cos(x)=c.

Guide to related sequences, with graphs included in Mathematica programs:

a.... b.... c.... x

1.... 1.... 1.... A199170, A199171

1.... 1.... 2.... A199172, A199173

1.... 1.... 3.... A199174, A199175

1.... 2.... 1.... A199176, A199177

1.... 2.... 2.... A199178, A199179

1.... 2.... 3.... A199180, A199181

1.... 3.... 1.... A199182, A199183

1.... 3.... 2.... A199184, A199185

1.... 3.... 3.... A199186, A199187

2.... 1.... 1.... A199188, A199189

2.... 1.... 2.... A199265, A199266

2.... 1.... 3.... A199267, A199268

2.... 2.... 1.... A199269, A199270

2.... 2.... 3.... A199271, A199272

2.... 3.... 1.... A199273, A199274

2.... 3.... 2.... A199275, A199276

2.... 3.... 3.... A199277, A199278

3.... 1.... 1.... A199279, A199280

3.... 1.... 2.... A199281, A199282

3.... 1.... 3.... A199283, A199284

3.... 2.... 1.... A199285, A199286

3.... 2.... 2.... A199287, A199288

3.... 2.... 3.... A199289, A199290

3.... 3.... 1.... A199291, A199292

3.... 3.... 2.... A199293, A199294

Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.

For an example related to A199170, take f(x,u,v)=x^2+u*xcos(x)-v and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

negative: -1.19835984451866026826502160343030898927268...

positive: 0.685174133854503187895211530638458709591...

MATHEMATICA

(* Program 1: A199170 and A199171 *)

a = 1; b = 1; c = 1;

f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c

Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]

RealDigits[r] (* A199170 *)