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%I #11 Apr 10 2021 15:39:46
%S 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,
%T 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,
%U 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
%N Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.
%C 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:
%C a.... c.... x
%C 1.... 1.... A196816
%C 1.... 2.... A201397
%C 1.... 3.... A201398
%C 1.... 4.... A201399
%C 1.... 5.... A201400
%C 1.... 6.... A201401
%C 1.... 7.... A201402
%C 1.... 8.... A201403
%C 1.... 9.... A201404
%C 1.... 10... A201405
%C 2.... 0.... A201406, A201407
%C 3.... 0.... A201408, A201409
%C 4.... 0.... A201410, A201411
%C 5.... 0.... A201412, A201413
%C 6.... 0.... A201414, A201415
%C 7.... 0.... A201416, A201417
%C 8.... 0.... A201418, A201419
%C 9.... 0.... A201420, A201421
%C 10... 0.... A201422, A201423
%C 3... -1.... A201515, A201516
%C 4... -1.... A201517, A201518
%C 5... -1.... A201519, A201520
%C 6... -1.... A201521, A201522
%C 7... -1.... A201523, A201524
%C 8... -1.... A201525, A201526
%C 9... -1.... A201527, A201528
%C 10.. -1.... A201529, A201530
%C 2.... 3.... A201531
%C 3.... 2.... A200619
%C 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.
%C 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.
%e x=1.2954596464154787686299132707186415897672...
%t (* Program 1: A201397 *)
%t a = 1; c = 2;
%t f[x_] := a*x^2 + c; g[x_] := Sec[x]
%t Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t RealDigits[r] (* A201397 *)
%t (* Program 2: implicit surface of u*x^2+v=sec(x) *)
%t Remove["Global`*"];
%t f[{x_, u_, v_}] := u*x^2 + v - Sec[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
%t ListPlot3D[Flatten[t, 1]] (* for A201397 *)
%Y Cf. A201280, A200614.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Dec 01 2011
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