A199265 -id:A199265

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A199170

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

+10
54

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

(list; constant; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Nov 03 2011

STATUS

approved

A199266

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

See A199170 for a guide to related sequences. The Mathematica program includes a graph.

LINKS

Table of n, a(n) for n=0..98.

EXAMPLE

negative: -1.11586547417780092002590535253585579273...

positive: 0.84824964906603356449300167136536010515870...

MATHEMATICA

a = 2; b = 1; c = 2;

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

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

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

RealDigits[r] (* A199265 *)

r = x /. FindRoot[f[x] == g[x], {x, .84, .85}, WorkingPrecision -> 110]

RealDigits[r] (* A199266 *)