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A199170
Decimal expansion of x<0 satisfying x^2+x*cos(x)=1.
54
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, 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, 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
OFFSET
1,3
COMMENTS
For many choices of a,b,c, there are exactly two numbers x satisfying a*x^2+b*x*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 1.... 1.... A199170, A199171
1.... 1.... 2.... A199172, A199173
1.... 1.... 3.... A199174, A199175
1.... 2.... 1.... A199176, A199177
1.... 2.... 2.... A199178, A199179
1.... 2.... 3.... A199180, A199181
1.... 3.... 1.... A199182, A199183
1.... 3.... 2.... A199184, A199185
1.... 3.... 3.... A199186, A199187
2.... 1.... 1.... A199188, A199189
2.... 1.... 2.... A199265, A199266
2.... 1.... 3.... A199267, A199268
2.... 2.... 1.... A199269, A199270
2.... 2.... 3.... A199271, A199272
2.... 3.... 1.... A199273, A199274
2.... 3.... 2.... A199275, A199276
2.... 3.... 3.... A199277, A199278
3.... 1.... 1.... A199279, A199280
3.... 1.... 2.... A199281, A199282
3.... 1.... 3.... A199283, A199284
3.... 2.... 1.... A199285, A199286
3.... 2.... 2.... A199287, A199288
3.... 2.... 3.... A199289, A199290
3.... 3.... 1.... A199291, A199292
3.... 3.... 2.... A199293, A199294
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.
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.
EXAMPLE
negative: -1.19835984451866026826502160343030898927268...
positive: 0.685174133854503187895211530638458709591...
MATHEMATICA
(* Program 1: A199170 and A199171 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
RealDigits[r] (* A199170 *)
r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
RealDigits[r] (* A199171 *)
(* Program 2: implicit surface of x^2+u*x*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0,
1.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]] (* for A199170 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 03 2011
STATUS
approved