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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 (list; constant; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 19 18:35 EDT 2022. Contains 353847 sequences. (Running on oeis4.)