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%I #9 Mar 30 2012 18:57:57
%S 1,1,9,8,3,5,9,8,4,4,5,1,8,6,6,0,2,6,8,2,6,5,0,2,1,6,0,3,4,3,0,3,0,8,
%T 9,8,9,2,7,2,6,8,0,9,3,5,8,7,4,8,2,5,6,9,0,1,4,4,4,9,2,3,8,6,8,6,4,2,
%U 7,1,7,6,1,4,9,7,1,9,1,2,5,5,9,1,7,1,4,2,8,9,1,6,9,7,2,0,9,5,4
%N Decimal expansion of x<0 satisfying x^2+x*cos(x)=1.
%C For many choices of a,b,c, there are exactly two numbers x satisfying a*x^2+b*x*cos(x)=c.
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c.... x
%C 1.... 1.... 1.... A199170, A199171
%C 1.... 1.... 2.... A199172, A199173
%C 1.... 1.... 3.... A199174, A199175
%C 1.... 2.... 1.... A199176, A199177
%C 1.... 2.... 2.... A199178, A199179
%C 1.... 2.... 3.... A199180, A199181
%C 1.... 3.... 1.... A199182, A199183
%C 1.... 3.... 2.... A199184, A199185
%C 1.... 3.... 3.... A199186, A199187
%C 2.... 1.... 1.... A199188, A199189
%C 2.... 1.... 2.... A199265, A199266
%C 2.... 1.... 3.... A199267, A199268
%C 2.... 2.... 1.... A199269, A199270
%C 2.... 2.... 3.... A199271, A199272
%C 2.... 3.... 1.... A199273, A199274
%C 2.... 3.... 2.... A199275, A199276
%C 2.... 3.... 3.... A199277, A199278
%C 3.... 1.... 1.... A199279, A199280
%C 3.... 1.... 2.... A199281, A199282
%C 3.... 1.... 3.... A199283, A199284
%C 3.... 2.... 1.... A199285, A199286
%C 3.... 2.... 2.... A199287, A199288
%C 3.... 2.... 3.... A199289, A199290
%C 3.... 3.... 1.... A199291, A199292
%C 3.... 3.... 2.... A199293, A199294
%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 A199170, take f(x,u,v)=x^2+u*xcos(x)-v 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 negative: -1.19835984451866026826502160343030898927268...
%e positive: 0.685174133854503187895211530638458709591...
%t (* Program 1: A199170 and A199171 *)
%t a = 1; b = 1; c = 1;
%t f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c
%t Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
%t RealDigits[r] (* A199170 *)
%t r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
%t RealDigits[r] (* A199171 *)
%t (* Program 2: implicit surface of x^2+u*x*cos(x)=v *)
%t f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0,
%t 1.9}, {v, u, 600}];
%t ListPlot3D[Flatten[t, 1]] (* for A199170 *)
%Y Cf. A199171, A197737, A198414, A198755, A198866.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Nov 03 2011
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