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A200338 Decimal expansion of least x>0 satisfying x^2+1=tan(x). 159
1, 1, 7, 2, 0, 9, 3, 6, 1, 7, 2, 8, 5, 6, 6, 9, 0, 3, 9, 6, 8, 7, 8, 1, 8, 7, 9, 5, 8, 1, 0, 8, 9, 8, 8, 0, 4, 0, 2, 4, 2, 4, 5, 7, 0, 8, 8, 0, 2, 7, 6, 3, 7, 1, 7, 6, 0, 1, 8, 6, 6, 3, 6, 7, 1, 2, 1, 8, 6, 6, 3, 4, 6, 0, 7, 6, 4, 1, 2, 2, 8, 3, 6, 5, 4, 5, 6, 1, 1, 2, 2, 8, 6, 7, 2, 3, 0, 3, 2 (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 satisfying a*x^2+b*x+c=tan(x) and 0<x<pi/2.

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

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

1.... 0.... 1.... A200338

1.... 0.... 2.... A200339

1.... 0.... 3.... A200340

1.... 0.... 4.... A200341

1.... 1.... 1.... A200342

1.... 1.... 2.... A200343

1.... 1.... 3.... A200344

1.... 1.... 4.... A200345

1.... 2.... 1.... A200346

1.... 2.... 2.... A200347

1.... 2.... 3.... A200348

1.... 2.... 4.... A200349

1.... 3.... 1.... A200350

1.... 3.... 2.... A200351

1.... 3.... 3.... A200352

1.... 3.... 4.... A200353

1.... 4.... 1.... A200354

1.... 4.... 2.... A200355

1.... 4.... 3.... A200356

1.... 4.... 4.... A200357

2.... 0.... 1.... A200358

2.... 0.... 3.... A200359

2.... 1.... 1.... A200360

2.... 1.... 2.... A200361

2.... 1.... 3.... A200362

2.... 1.... 4.... A200363

2.... 2.... 1.... A200364

2.... 2.... 3.... A200365

2.... 3.... 1.... A200366

2.... 3.... 2.... A200367

2.... 3.... 3.... A200368

2.... 3.... 4.... A200369

2.... 4.... 1.... A200382

2.... 4.... 3.... A200383

3.... 0.... 1.... A200384

3.... 0.... 2.... A200385

3.... 0.... 4.... A200386

3.... 1.... 1.... A200387

3.... 1.... 2.... A200388

3.... 1.... 3.... A200389

3.... 1.... 4.... A200390

3.... 2.... 1.... A200391

3.... 2.... 2.... A200392

3.... 2.... 3.... A200393

3.... 2.... 4.... A200394

3.... 3.... 1.... A200395

3.... 3.... 2.... A200396

3.... 3.... 4.... A200397

3.... 4.... 1.... A200398

3.... 4.... 2.... A200399

3.... 4.... 3.... A200400

3.... 4.... 4.... A200401

4.... 0.... 1.... A200410

4.... 0.... 3.... A200411

4.... 1.... 1.... A200412

4.... 1.... 2.... A200413

4.... 1.... 3.... A200414

4.... 1.... 4.... A200415

4.... 2.... 1.... A200416

4.... 2.... 3.... A200417

4.... 3.... 1.... A200418

4.... 3.... 2.... A200419

4.... 3.... 3.... A200420

4.... 3.... 4.... A200421

4.... 4.... 1.... A200422

4.... 4.... 3.... A200423

1... -1.... 1.... A200477

1... -1.... 2.... A200478

1... -1.... 3.... A200479

1... -1.... 4.... A200480

1... -2.... 1.... A200481

1... -2.... 2.... A200482

1... -2.... 3.... A200483

1... -2.... 4.... A200484

1... -3.... 1.... A200485

1... -3.... 2.... A200486

1... -3.... 3.... A200487

1... -3.... 4.... A200488

1... -4.... 1.... A200489

1... -4.... 2.... A200490

1... -4.... 3.... A200491

1... -4.... 4.... A200492

2... -1.... 1.... A200493

2... -1.... 2.... A200494

2... -1.... 3.... A200495

2... -1.... 4.... A200496

2... -2.... 1.... A200497

2... -2.... 3.... A200498

2... -3.... 1.... A200499

2... -3.... 2.... A200500

2... -3.... 3.... A200501

2... -3.... 4.... A200502

2... -4.... 1.... A200584

2... -4.... 3.... A200585

2... -1.... 2.... A200586

2... -1.... 3.... A200587

2... -1.... 4.... A200588

3... -2.... 1.... A200589

3... -2.... 2.... A200590

3... -2.... 3.... A200591

3... -2.... 4.... A200592

3... -3.... 1.... A200593

3... -3.... 2.... A200594

3... -3.... 4.... A200595

3... -4.... 1.... A200596

3... -4.... 2.... A200597

3... -4.... 3.... A200598

3... -4.... 4.... A200599

4... -1.... 1.... A200600

4... -1.... 2.... A200601

4... -1.... 3.... A200602

4... -1.... 4.... A200603

4... -2.... 1.... A200604

4... -2.... 3.... A200605

4... -3.... 1.... A200606

4... -3.... 2.... A200607

4... -3.... 3.... A200608

4... -3.... 4.... A200609

4... -4.... 1.... A200610

4... -4.... 3.... A200611

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 A200338, take f(x,u,v)=x^2+u*x+v-tan(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=1.17209361728566903968781879581089880...

MATHEMATICA

(* Program 1:  A200338 *)

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

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

Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]

RealDigits[r]  (* A200338 *)

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

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

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.57}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];

ListPlot3D[Flatten[t, 1]]  (* for A200388 *)

CROSSREFS

Cf. A197737, A198414, A198755, A198866, A199170, A199370, A199429, A199597, A199949.

Sequence in context: A093954 A177703 A266814 * A153589 A010505 A020844

Adjacent sequences:  A200335 A200336 A200337 * A200339 A200340 A200341

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Nov 16 2011

STATUS

approved

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Last modified April 20 16:17 EDT 2019. Contains 322310 sequences. (Running on oeis4.)