A199618 -id:A199618

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A199597

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

+10
99

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

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

OFFSET

1,3

COMMENTS

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

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

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

1.... 1.... 2.... A199597

1.... 1.... 3.... A199598

1.... 1.... 4.... A199599

1.... 2.... 1.... A199600

1.... 2.... 3.... A199601

1.... 2.... 4.... A199602

1.... 3.... 0.... A199603, A199604

1.... 3.... 1.... A199605, A199606

1.... 3.... 2.... A199607, A199608

1.... 3.... 3.... A199609, A199610

1.... 4.... 0.... A199611, A199612

1.... 4.... 1.... A199613, A199614

1.... 4.... 2.... A199615, A199616

1.... 4.... 3.... A199617, A199618

1.... 4.... 4.... A199619, A199620

2.... 1.... 0.... A199621

2.... 1.... 2.... A199622

2.... 1.... 3.... A199623

2.... 1.... 4.... A199624

2.... 2.... 1.... A199625

2.... 2.... 3.... A199661

3.... 1.... 0.... A199662

3.... 1.... 2.... A199663

3.... 1.... 3.... A199664

3.... 1.... 4.... A199665

3.... 2.... 0.... A199666

3.... 2.... 1.... A199667

3.... 2.... 3.... A199668

3.... 2.... 4.... A199669

1... -1.... 0.... A003957

1... -1.... 1.... A199722

1... -1.... 2.... A199721

1... -1.... 3.... A199720

1... -1.... 4.... A199719

1... -2.... 1.... A199726

1... -2.... 2.... A199725

1... -2.... 3.... A199724

1... -2.... 4.... A199723

1... -3.... 1.... A199730

1... -3.... 2.... A199729

1... -3.... 3.... A199728

1... -3.... 4.... A199727

1... -4.... 1.... A199737. A199738

1... -4.... 2.... A199735, A199736

1... -4.... 3.... A199733, A199734

1... -4.... 4.... A199731. A199732

2... -1.... 1.... A199742

2... -1.... 2.... A199741

2... -1.... 3.... A199740

2... -1.... 4.... A199739

2... -2.... 1.... A199776

2... -2.... 3.... A199775

2... -3.... 1.... A199780

2... -3.... 2.... A199779

2... -3.... 3.... A199778

2... -3.... 4.... A199777

2... -4.... 1.... A199782

2... -4.... 3.... A199781

3... -4.... 1.... A199786

3... -4.... 2.... A199785

3... -4.... 3.... A199784

3... -4.... 4.... A199783

3... -3.... 1.... A199789

3... -3.... 2.... A199788

3... -3.... 4.... A199787

3... -2.... 1.... A199793

3... -2.... 2.... A199792

3... -2.... 3.... A199791

3... -2.... 4.... A199790

3... -1.... 1.... A199797

3... -1.... 2.... A199796

3... -1.... 3.... A199795

3... -1.... 4.... A199794

4... -4.... 1.... A199873

4... -4.... 3.... A199872

4... -3.... 1.... A199871

4... -3.... 2.... A199870

4... -3.... 3.... A199869

4... -3.... 4.... A199868

4... -2.... 1.... A199867

4... -2.... 3.... A199866

4... -1.... 1.... A199865

4... -1.... 2.... A199864

4... -1.... 3.... A199863

4... -1.... 4.... A199862

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 A199597, take f(x,u,v)=x^2+u*x*cos(x)-v*sin(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..100.

EXAMPLE

1.1881851344514388032178109729076525973...

MATHEMATICA

(* Program 1: A199597 *)

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

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

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

r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]

RealDigits[r] (* A199597 *)

(* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)

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

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];

ListPlot3D[Flatten[t, 1]] (* for A199597 *)

CROSSREFS

Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Nov 08 2011

EXTENSIONS

Edited by Georg Fischer, Aug 03 2021

STATUS

approved

A199617

Decimal expansion of least x satisfying x^2+4*x*cos(x)=3*sin(x).

+10
3

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

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

OFFSET

0,1

COMMENTS

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

LINKS

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

EXAMPLE

least: -0.5535433817860336287020344905911816930...

greatest: 3.4822676247861932090867036675576803...

MATHEMATICA

a = 1; b = 4; c = 3;

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

Plot[{f[x], g[x]}, {x, -2, 4}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, -.56, -.55}, WorkingPrecision -> 110]

RealDigits[r] (* A199617, least of 4 roots *)

r = x /. FindRoot[f[x] == g[x], {x, 3.4, 3.5}, WorkingPrecision -> 110]

RealDigits[r] (* A199618, greatest of 4 roots *)