A198414 Decimal expansion of x > 0 satisfying x^2 = 2*sin(x).

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

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.....A198418

1.....1.....4.....A198419

1.....2.....1.....A198424

1.....2.....3.....A198425

1.....2.....4.....A198426

1....-1.....1.....A198420

1....-1.....1.....A198420

1....-1.....2.....A198421

1....-1.....3.....A198422

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

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 * A309721 A110854 A278086

Adjacent sequences: A198411 A198412 A198413 * A198415 A198416 A198417

Keyword

nonn,cons

Author

Clark Kimberling, Oct 24 2011

Extensions

Edited by Georg Fischer, Aug 01 2021

Status

approved