A200614 - OEIS

%I #23 Jan 30 2025 17:43:51

%S 8,3,9,5,8,2,2,5,9,0,4,5,3,0,2,9,4,1,5,1,3,7,6,4,0,0,8,8,6,3,8,0,4,9,

%T 8,6,3,0,8,4,1,6,5,4,1,0,2,6,9,4,4,0,9,0,0,3,3,4,9,1,4,3,4,0,6,7,6,5,

%U 8,4,1,4,6,1,0,4,1,1,6,7,4,2,5,9,5,3,5,3,0,0,2,3,6,6,2,4,6,0,5

%N Decimal expansion of the lesser of two values of x satisfying 3*x^2 - 1 = tan(x) and 0 < x < Pi/2.

%C For many choices of a and c, there are exactly two values of x satisfying a*x^2 - c = tan(x) and 0 < x < Pi/2; for other choices, there is exactly once such value.

%C Guide to related sequences, with graphs included in Mathematica programs:

%C a.... c.... x

%C 3.... 1.... A200614, A200615

%C 4.... 1.... A200616, A200617

%C 5.... 1.... A200620, A200621

%C 5.... 2.... A200622, A200623

%C 5.... 3.... A200624, A200625

%C 5.... 4.... A200626, A200627

%C 5... -1.... A200628

%C 5... -2.... A200629

%C 5... -3.... A200630

%C 5... -4.... A200631

%C 6.... 1.... A200633, A200634

%C 6.... 5.... A200635, A200636

%C 6... -1.... A200637

%C 6... -5.... A200638

%C 1... -5.... A200639

%C 2... -5.... A200640

%C 3... -5.... A200641

%C 4... -5.... A200642

%C 2.... 0.... A200679, A200680

%C 3.... 0.... A200681, A200682

%C 4.... 0.... A200683, A200684

%C 5.... 0.... A200618

%C 6.... 0.... A200632

%C 7.... 0.... A200643

%C 8.... 0.... A200644

%C 9.... 0.... A200645

%C 10... 0.... A200646

%C -1... 1.... A200685

%C -1... 2.... A200686

%C -1... 3.... A200687

%C -1... 4.... A200688

%C -1... 5.... A200689

%C -1... 6.... A200690

%C -1... 7.... A200691

%C -1... 8.... A200692

%C -1... 9.... A200693

%C -1... 10... A200694

%C -2... 1.... A200695

%C -2... 3.... A200696

%C -3... 1.... A200697

%C -3... 2.... A200698

%C -4... 1.... A200699

%C -5... 1.... A200700

%C -6... 1.... A200701

%C -7... 1.... A200702

%C -8... 1.... A200703

%C -9... 1.... A200704

%C -10.. 1.... A200705

%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 A200614, take f(x,u,v) = u*x^2 - v = tan(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 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e lesser:  0.839582259045302941513764008863804986308...

%e greater: 1.350956593976519397744696294368524715373...

%t (* Program 1:  A200614 and A200615 *)

%t a = 3; c = 1;

%t f[x_] := a*x^2 - c; g[x_] := Tan[x]

%t Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]

%t RealDigits[r]   (* A200614 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]

%t RealDigits[r]   (* A200615 *)

%t (* Program 2: implicit surface of u*x^2-v=tan(x) *)

%t f[{x_, u_, v_}] := u*x^2 - v - Tan[x];

%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.55}]}, {u, 1, 20}, {v, -20, 0}];

%t ListPlot3D[Flatten[t, 1]]  (* for A200614 *)

%Y Cf. A200615, A200338.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Nov 20 2011