A198120 -id:A198120

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A197737

Decimal expansion of x<0 having x^2+x=cos(x).

+0
144

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

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

OFFSET

1,2

COMMENTS

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

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

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

1.... 0.... 1.... A125578

1.... 0.... 2.... A197806

1.... 0.... 3.... A197807

1.... 0.... 4.... A197808

1.... 1.... 1.... A197737, A197738

1.... 1.... 2.... A197809, A197810

1.... 1.... 3.... A197811, A197812

1.... 1.... 4.... A197813, A197814

1... -2... -1.... A197815, A197820

1... -3... -1.... A197825, A197831

1... -4... -1.... A197839, A197840

1.... 2.... 1.... A197841, A197842

1.... 2.... 2.... A197843, A197844

1.... 2.... 3.... A197845, A197846

1.... 2.... 4.... A197847, A197848

1... -2... -2.... A197849, A197850

1... -3... -2.... A198098, A198099

1... -4... -2.... A198100, A198101

1.... 3.... 1.... A198102, A198103

1.... 3.... 2.... A198104, A198105

1.... 3.... 3.... A198106, A198107

1.... 3.... 4.... A198108, A198109

1... -2... -3.... A198140, A198141

1... -3... -3.... A198142, A198143

1... -4... -3.... A198144, A198145

2.... 0.... 1.... A198110

2.... 0.... 3.... A198111

2.... 1.... 1.... A198112, A198113

2.... 1.... 2.... A198114, A198115

2.... 1.... 3.... A198116, A198117

2.... 1.... 4.... A198118, A198119

2.... 1... -1.... A198120, A198121

2... -4... -1.... A198122, A198123

2.... 2.... 1.... A198124, A198125

2.... 2.... 3.... A198126, A198127

2.... 3.... 1.... A198128, A198129

2.... 3.... 2.... A198130, A198131

2.... 3.... 3.... A198132, A198133

2.... 3.... 4.... A198134, A198135

2... -4... -3.... A198136, A198137

3.... 0.... 1.... A198211

3.... 0.... 2.... A198212

3.... 0.... 4.... A198213

3.... 1.... 1.... A198214, A198215

3.... 1.... 2.... A198216, A198217

3.... 1.... 3.... A198218, A198219

3.... 1.... 4.... A198220, A198221

3.... 2.... 1.... A198222, A198223

3.... 2.... 2.... A198224, A198225

3.... 2.... 3.... A198226, A198227

3.... 2.... 4.... A198228, A198229

3.... 3.... 1.... A198230, A198231

3.... 3.... 2.... A198232, A198233

3.... 3.... 4.... A198234, A198235

3.... 4.... 1.... A198236, A198237

3.... 4.... 2.... A198238, A198239

3.... 4.... 3.... A198240, A198241

3.... 4.... 4.... A198138, A198139

3... -4... -1.... A198345, A198346

4.... 0.... 1.... A198347

4.... 0.... 3.... A198348

4.... 1.... 1.... A198349, A198350

4.... 1.... 2.... A198351, A198352

4.... 1.... 3.... A198353, A198354

4.... 1.... 4.... A198355, A198356

4.... 2.... 1.... A198357, A198358

4.... 2.... 3.... A198359, A198360

4.... 3.... 1.... A198361, A198362

4.... 3.... 2.... A198363, A198364

4.... 3.... 3.... A198365, A198366

4.... 3.... 4.... A198367, A198368

4.... 4.... 1.... A198369, A198370

4.... 4.... 3.... A198371, A198372

4... -4... -1.... A198373, A198374

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 A197737, take f(x,u,v)=x^2+u*x-v*cos(x) 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: -1.25115183522076481159287006878816185994...

positive: 0.55000934992726156666495361947172926116...

MATHEMATICA

(* Program 1: A197738 *)

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

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

Plot[{f[x], g[x]}, {x, -2, 1}]

r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]

RealDigits[r1] (* A197737 *)

r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]

RealDigits[r1] (* A197738 *)

(* Program 2: implicit surface of x^2+u*x=v*cos(x) *)

f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x];

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}];

ListPlot3D[Flatten[t, 1]] (* for A197737 *)

PROG

(PARI) A197737_vec(N=150)={localprec(N+10); digits(solve(x=-1.5, -1, x^2+x-cos(x))\.1^N)} \\ M. F. Hasler, Aug 05 2021

CROSSREFS

Cf. A197738.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 20 2011

STATUS

approved

A198121

Decimal expansion of greatest x having 2*x^2-3x=-cos(x).

+0
3

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

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

OFFSET

1,2

COMMENTS

See A197737 for a guide to related sequences. The Mathematica program includes a graph.

LINKS

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

EXAMPLE

least x: 0.423418867436956390254901914567137...

greatest x: 1.4633628272964311451052964261613...

MATHEMATICA

a = 2; b = -3; c = -1;

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

Plot[{f[x], g[x]}, {x, -1, 2}]

r1 = x /. FindRoot[f[x] == g[x], {x, -.43, -.42}, WorkingPrecision -> 110]

RealDigits[r1](* A198120 *)

r2 = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]

RealDigits[r2](* A198121 *)