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A201280
Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.
46
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.
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 29 2011
EXTENSIONS
Edited and a(90) onwards corrected by Georg Fischer, Aug 03 2021
STATUS
approved