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A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1. 52
7, 2, 2, 5, 8, 7, 5, 4, 9, 9, 2, 2, 5, 2, 4, 7, 6, 8, 3, 5, 5, 9, 3, 2, 8, 7, 2, 8, 7, 7, 1, 9, 6, 7, 5, 5, 1, 5, 9, 6, 4, 5, 9, 2, 1, 1, 4, 3, 9, 4, 2, 6, 9, 8, 0, 7, 7, 6, 5, 1, 4, 7, 6, 0, 2, 5, 9, 0, 9, 4, 2, 5, 0, 7, 3, 1, 6, 0, 1, 8, 3, 0, 3, 4, 3, 5, 6, 2, 9, 4, 1, 8, 7, 2, 7, 9, 8, 3, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 1.... 1.... A199370
1.... 1.... 2.... A199371
1.... 1.... 3.... A199372
1.... 2.... 1.... A199373
1.... 2.... 2.... A199374
1.... 2.... 3.... A199375
1.... 3.... 1.... A199376
1.... 3.... 2.... A199377
1.... 3.... 3.... A199378
2.... 1.... 1.... A199379
2.... 1.... 2.... A199180
2.... 1.... 3.... A199181
2.... 2.... 1.... A199182
2.... 2.... 3.... A199183
2.... 3.... 1.... A199184
2.... 3.... 2.... A199185
2.... 3.... 3.... A199186
2.... 1.... 1.... A199187
3.... 1.... 2.... A199188
3.... 1.... 3.... A199189
3.... 2.... 1.... A199190
3.... 2.... 2.... A199191
3.... 2.... 3.... A199192
3.... 3.... 1.... A199193
3.... 3.... 2.... A199195
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 A199370, take f(x,u,v)=x^2+u*x*sin(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.
LINKS
EXAMPLE
x=0.722587549922524768355932872877196755159...
MATHEMATICA
(* Program 1: A199370 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -1, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .72, .73}, WorkingPrecision -> 110]
RealDigits[r] (* A199370 *)
(* Program 2: implicit surface of x^2+u*x*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 2.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]] (* for A199370 *)
CROSSREFS
Sequence in context: A010141 A316247 A299922 * A334961 A154759 A300304
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 05 2011
STATUS
approved

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Last modified February 26 17:57 EST 2024. Contains 370352 sequences. (Running on oeis4.)