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A201397
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Decimal expansion of x satisfying x^2+2=sec(x) and 0<x<pi.
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46
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..99.
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EXAMPLE
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x=1.2954596464154787686299132707186415897672...
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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Cf. A201280, A200614.
Sequence in context: A021776 A241995 A019708 * A077124 A051491 A253169
Adjacent sequences: A201394 A201395 A201396 * A201398 A201399 A201400
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KEYWORD
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nonn,cons
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AUTHOR
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Clark Kimberling, Dec 01 2011
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STATUS
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approved
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