A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.

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

Offset

1,2

Comments

For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = sec(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:

a.... c.... x

1.... 1.... A196816

1.... 2.... A201397

1.... 3.... A201398

1.... 4.... A201399

1.... 5.... A201400

1.... 6.... A201401

1.... 7.... A201402

1.... 8.... A201403

1.... 9.... A201404

1.... 10... A201405

2.... 0.... A201406, A201407

3.... 0.... A201408, A201409

4.... 0.... A201410, A201411

5.... 0.... A201412, A201413

6.... 0.... A201414, A201415

7.... 0.... A201416, A201417

8.... 0.... A201418, A201419

9.... 0.... A201420, A201421

10... 0.... A201422, A201423

3... -1.... A201515, A201516

4... -1.... A201517, A201518

5... -1.... A201519, A201520

6... -1.... A201521, A201522

7... -1.... A201523, A201524

8... -1.... A201525, A201526

9... -1.... A201527, A201528

10.. -1.... A201529, A201530

2.... 3.... A201531

3.... 2.... A200619

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 A201397, take f(x,u,v) = u*x^2 + v = sec(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.

Index entries for transcendental numbers.

Example

1.2954596464154787686299132707186415897672...

Mathematica

(* Program 1:  A201397 *)
a = 1; c = 2;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]    (* A201397 *)
(* Program 2: implicit surface of u*x^2+v=sec(x) *)
Remove["Global`*"];
f[{x_, u_, v_}] := u*x^2 + v - Sec[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
ListPlot3D[Flatten[t, 1]]  (* for A201397 *)

Crossrefs

Cf. A201280, A200614.

Sequence in context: A021776 A241995 A019708 * A077124 A051491 A339204

Adjacent sequences: A201394 A201395 A201396 * A201398 A201399 A201400

Keyword

nonn,cons

Author

Clark Kimberling, Dec 01 2011

Status

approved