A201280 Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.

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

Offset

0,1

Comments

For many choices of a and c, there is exactly one x satisfying a*x^2 + c = cot(x) and 0 < x < Pi.

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

a.... c.... x

1.... 1.... A201280

1.... 2.... A201281

1.... 3.... A201282

1.... 4.... A201283

1.... 5.... A201284

1.... 6.... A201285

1.... 7.... A201286

1.... 8.... A201287

1.... 9.... A201288

1.... 10... A201289

1.... 0.... A201294

1... -1.... A201295

1... -2.... A201296

1... -3.... A201297

1... -4.... A201298

1... -5.... A201299

1... -6.... A201315

1... -7.... A201316

1... -8.... A201317

1... -9.... A201318

1.. -10.... A201319

2.... 0.... A201329

3.... 0.... A201330

4.... 0.... A201331

5.... 0.... A201332

6.... 0.... A201333

7.... 0.... A201334

8.... 0.... A201335

9.... 0.... A201336

10... 0.... A201337

2... -1.... A201320

3... -1.... A201321

4... -1.... A201322

5... -1.... A201323

6... -1.... A201324

7... -1.... A201325

8... -1.... A201326

9... -1.... A201327

10.. -1.... A201328

2.... 1.... A201290

2.... 3.... A201291

2... -3.... A201394

3.... 1.... A201292

3.... 2.... A201293

3... -2.... A201395

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 A201280, take f(x,u,v) = u*x^2 - v - cot(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=0..98.

Example

0.62389956058090344363990329394632442...

Mathematica

(* Program 1: A201280 *)
a = 1; c = 1;
f[x_] := a*x^2 + c; g[x_] := Cot[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .62, .63}, WorkingPrecision -> 110]
RealDigits[r]   (* A201280 *)
(* Program 2: implicit surface of u*x^2-v=cot(x) *)
f[{x_, u_, v_}] := u*x^2 - v - Cot[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .001, Pi}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
ListPlot3D[Flatten[t, 1]]  (* for A201280 *)

Crossrefs

Cf. A200614.

Sequence in context: A195453 A259543 A256632 * A248274 A267568 A182011

Adjacent sequences: A201277 A201278 A201279 * A201281 A201282 A201283

Keyword

nonn,cons

Author

Clark Kimberling, Nov 29 2011

Extensions

Edited and a(90) onwards corrected by Georg Fischer, Aug 03 2021

Status

approved