A201335 -id:A201335 - cp's OEIS frontend

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

Clark Kimberling, Nov 29 2011

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.... A201280
... 2.... A201281
... 3.... A201282
... 4.... A201283
... 5.... A201284
... 6.... A201285
... 7.... A201286
... 8.... A201287
... 9.... A201288
... 10... A201289
... 0.... A201294
.. -1.... A201295
.. -2.... A201296
.. -3.... A201297
.. -4.... A201298
.. -5.... A201299
.. -6.... A201315
.. -7.... A201316
.. -8.... A201317
.. -9.... A201318
. -10.... A201319
... 0.... A201329
... 0.... A201330
... 0.... A201331
... 0.... A201332
... 0.... A201333
... 0.... A201334
... 0.... A201335
... 0.... A201336
.. 0.... A201337
.. -1.... A201320
.. -1.... A201321
.. -1.... A201322
.. -1.... A201323
.. -1.... A201324
.. -1.... A201325
.. -1.... A201326
.. -1.... A201327
. -1.... A201328
... 1.... A201290
... 3.... A201291
.. -3.... A201394
... 1.... A201292
... 2.... A201293
.. -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.

			0.62389956058090344363990329394632442...

Index entries for transcendental numbers.

Cf. A200614.

(* 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 *)

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

cp's OEIS Frontend

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

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