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%I #16 Apr 09 2021 14:52:28
%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.... A200239
%C 2... -5.... A200240
%C 3... -5.... A200241
%C 4... -5.... A200242
%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.
%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
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