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

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 *)
r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
RealDigits[r]  (* A199171 *)
(* Program 2: implicit surface of x^2+u*x*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0,
    1.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]]  (* for A199170 *)

Crossrefs

Cf. A199171, A197737, A198414, A198755, A198866.

Sequence in context: A363633 A224236 A118427 * A155532 A086306 A094139

Adjacent sequences: A199167 A199168 A199169 * A199171 A199172 A199173

Keyword

nonn,cons

Author

Clark Kimberling, Nov 03 2011

Status

approved