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 A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi. 46
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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. EXAMPLE x=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

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Last modified December 3 18:21 EST 2023. Contains 367540 sequences. (Running on oeis4.)