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A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x). 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.

Sequence in context: A265308 A319858 A351210 * A197848 A224875 A242588

Adjacent sequences:  A199594 A199595 A199596 * A199598 A199599 A199600

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Nov 08 2011

EXTENSIONS

Edited by Georg Fischer, Aug 03 2021

STATUS

approved

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Last modified June 26 14:05 EDT 2022. Contains 354884 sequences. (Running on oeis4.)