login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A198414 Decimal expansion of x>0 satisfying x^2=2*sin(x). 107
1, 4, 0, 4, 4, 1, 4, 8, 2, 4, 0, 9, 2, 4, 3, 4, 3, 6, 4, 1, 4, 8, 3, 2, 7, 9, 4, 3, 7, 4, 5, 7, 5, 8, 6, 0, 3, 7, 2, 5, 7, 1, 6, 1, 3, 7, 0, 4, 9, 1, 1, 4, 8, 1, 0, 9, 4, 4, 8, 2, 4, 3, 5, 4, 8, 7, 7, 5, 2, 5, 2, 9, 5, 6, 1, 7, 1, 4, 4, 3, 6, 2, 1, 2, 0, 5, 1, 0, 1, 5, 2, 4, 8, 2, 0, 8, 1, 7, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For many choices of a,b,c, there is a unique nonzero number x satisfying a*x^2+b*x=c*sin(x).

Specifically, for a>0 and many choices of b and c, the curves y=ax^2+bx and y=c*sin(x) meet in a single point if and only if b=c, in which case the curves have a common tangent line, y=c*x.  If b<c, the curves meet in quadrant 1; if b>c, they meet in quadrant 2.

Guide to related sequences (with graphs included in Mathematica programs):

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

1.....0.....1.....A124597

1.....0.....2.....A198414

1.....0.....3.....A198415

1.....0.....4.....A198416

1.....1.....2.....A198417

1.....1.....3.....A197418

1.....1.....4.....A197419

1.....2.....1.....A197424

1.....2.....3.....A197425

1.....2.....4.....A197426

1....-1.....1.....A197420

1....-1.....1.....A197420

1....-1.....2.....A197421

1....-1.....3.....A197422

1....-2.....1.....A198427

1....-2.....2.....A198428

1....-2.....3.....A198429

1....-2.....4.....A198430

1....-3.....1.....A198431

1....-3.....2.....A198432

1....-3.....3.....A198433

1....-3.....4.....A198488

1....-4.....1.....A198489

1....-4.....2.....A198490

1....-4.....3.....A198491

1....-4.....4.....A198492

2.....0.....1.....A198583

2.....0.....3.....A198605

2.....1.....2.....A198493

2.....1.....3.....A198494

2.....1.....4.....A198495

2.....2.....1.....A198496

2.....2.....3.....A198497

2.....3.....1.....A198608

2.....3.....2.....A198609

2.....3.....4.....A198610

2.....4.....1.....A198611

2.....4.....3.....A198612

2....-1.....1.....A198546

2....-1.....2.....A198547

2....-1.....3.....A198548

2....-1.....4.....A198549

2....-2.....3.....A198559

2....-3.....1.....A198566

2....-3.....2.....A198567

2....-3.....3.....A198568

2....-3.....4.....A198569

2....-4.....1.....A198577

2....-4.....3.....A198578

3.....0.....1.....A198501

3.....0.....2.....A198502

3.....1.....2.....A198498

3.....1.....3.....A198499

3.....1.....4.....A198500

3.....2.....1.....A198613

3.....2.....3.....A198614

3.....2.....4.....A198615

3.....3.....1.....A198616

3.....3.....2.....A198617

3.....3.....4.....A198618

3.....4.....1.....A198606

3.....4.....2.....A198607

3.....4.....3.....A198619

3....-1.....1.....A198550

3....-1.....2.....A198551

3....-1.....3.....A198552

3....-1.....4.....A198553

3....-2.....1.....A198560

3....-2.....2.....A198561

3....-2.....3.....A198562

3....-2.....4.....A198563

3....-3.....1.....A198570

3....-3.....2.....A198571

3....-3.....4.....A198572

3....-4.....1.....A198579

3....-4.....2.....A198580

3....-4.....3.....A198581

3....-4.....4.....A198582

4.....0.....1.....A198503

4.....0.....3.....A198504

4.....1.....2.....A198505

4.....1.....3.....A198506

4.....1.....4.....A198507

4.....2.....1.....A198539

4.....2.....3.....A198540

4.....3.....1.....A198541

4.....3.....2.....A198542

4.....3.....4.....A198543

4.....4.....1.....A198544

4.....4.....3.....A198545

4....-1.....1.....A198554

4....-1.....2.....A198555

4....-1.....3.....A198556

4....-1.....4.....A198557

4....-1.....1.....A198554

4....-2.....1.....A198564

4....-2.....3.....A198565

4....-3.....1.....A198573

4....-3.....2.....A198574

4....-3.....3.....A198575

4....-3.....4.....A198576

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 A198414, take f(x,u,v)=x^2+u*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..99.

EXAMPLE

x=1.4044148240924343641483279437457586037...

MATHEMATICA

(* Program 1: A198414 *)

a = 1; b = 0; c = 2;

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

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

r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.41}, WorkingPrecision -> 110]

RealDigits[r] (* A198414 *)

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

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

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .01, 6}]}, {u, .1, 100}, {v, u, 100}];

ListPlot3D[Flatten[t, 1]]

CROSSREFS

Cf. A197737.

Sequence in context: A062524 A152856 A031362 * A110854 A278086 A021716

Adjacent sequences:  A198411 A198412 A198413 * A198415 A198416 A198417

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 24 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 24 02:32 EDT 2019. Contains 326260 sequences. (Running on oeis4.)