cp's OEIS Frontend

A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.

%I A201397 #13 Jan 30 2025 15:31:52
%S A201397 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 A201397 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 A201397 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 A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.
%C A201397 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 A201397 a.... c.... x
%C A201397 1.... 1.... A196816
%C A201397 1.... 2.... A201397
%C A201397 1.... 3.... A201398
%C A201397 1.... 4.... A201399
%C A201397 1.... 5.... A201400
%C A201397 1.... 6.... A201401
%C A201397 1.... 7.... A201402
%C A201397 1.... 8.... A201403
%C A201397 1.... 9.... A201404
%C A201397 1.... 10... A201405
%C A201397 2.... 0.... A201406, A201407
%C A201397 3.... 0.... A201408, A201409
%C A201397 4.... 0.... A201410, A201411
%C A201397 5.... 0.... A201412, A201413
%C A201397 6.... 0.... A201414, A201415
%C A201397 7.... 0.... A201416, A201417
%C A201397 8.... 0.... A201418, A201419
%C A201397 9.... 0.... A201420, A201421
%C A201397 10... 0.... A201422, A201423
%C A201397 3... -1.... A201515, A201516
%C A201397 4... -1.... A201517, A201518
%C A201397 5... -1.... A201519, A201520
%C A201397 6... -1.... A201521, A201522
%C A201397 7... -1.... A201523, A201524
%C A201397 8... -1.... A201525, A201526
%C A201397 9... -1.... A201527, A201528
%C A201397 10.. -1.... A201529, A201530
%C A201397 2.... 3.... A201531
%C A201397 3.... 2.... A200619
%C A201397 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 A201397 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.
%H A201397 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201397 1.2954596464154787686299132707186415897672...
%t A201397 (* Program 1:  A201397 *)
%t A201397 a = 1; c = 2;
%t A201397 f[x_] := a*x^2 + c; g[x_] := Sec[x]
%t A201397 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201397 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A201397 RealDigits[r]    (* A201397 *)
%t A201397 (* Program 2: implicit surface of u*x^2+v=sec(x) *)
%t A201397 Remove["Global`*"];
%t A201397 f[{x_, u_, v_}] := u*x^2 + v - Sec[x];
%t A201397 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
%t A201397 ListPlot3D[Flatten[t, 1]]  (* for A201397 *)
%Y A201397 Cf. A201280, A200614.
%K A201397 nonn,cons
%O A201397 1,2
%A A201397 _Clark Kimberling_, Dec 01 2011