A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.

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

Offset

0,1

Comments

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c.

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

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

1.... 1.... 1.... A199370

1.... 1.... 2.... A199371

1.... 1.... 3.... A199372

1.... 2.... 1.... A199373

1.... 2.... 2.... A199374

1.... 2.... 3.... A199375

1.... 3.... 1.... A199376

1.... 3.... 2.... A199377

1.... 3.... 3.... A199378

2.... 1.... 1.... A199379

2.... 1.... 2.... A199180

2.... 1.... 3.... A199181

2.... 2.... 1.... A199182

2.... 2.... 3.... A199183

2.... 3.... 1.... A199184

2.... 3.... 2.... A199185

2.... 3.... 3.... A199186

2.... 1.... 1.... A199187

3.... 1.... 2.... A199188

3.... 1.... 3.... A199189

3.... 2.... 1.... A199190

3.... 2.... 2.... A199191

3.... 2.... 3.... A199192

3.... 3.... 1.... A199193

3.... 3.... 2.... A199195

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 A199370, take f(x,u,v)=x^2+u*x*sin(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=0..98.

Example

x=0.722587549922524768355932872877196755159...

Mathematica

(* Program 1: A199370 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -1, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .72, .73}, WorkingPrecision -> 110]
RealDigits[r]  (* A199370 *)
(* Program 2: implicit surface of x^2+u*x*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 2.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]]  (* for A199370 *)

Crossrefs

Cf. A199371, A199170, A198866, A198755, A198414, A197737.

Sequence in context: A010141 A316247 A299922 * A334961 A154759 A300304

Adjacent sequences: A199367 A199368 A199369 * A199371 A199372 A199373

Keyword

nonn,cons

Author

Clark Kimberling, Nov 05 2011

Status

approved