A197737 Decimal expansion of x<0 having x^2+x=cos(x).

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

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.

Sequence in context: A092134 A181779 A024548 * A189824 A197814 A091772

Adjacent sequences: A197734 A197735 A197736 * A197738 A197739 A197740

Keyword

nonn,cons

Author

Clark Kimberling, Oct 20 2011

Status

approved