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A198866
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Decimal expansion of x < 0 satisfying x^2 + sin(x) = 1.
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57
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1, 4, 0, 9, 6, 2, 4, 0, 0, 4, 0, 0, 2, 5, 9, 6, 2, 4, 9, 2, 3, 5, 5, 9, 3, 9, 7, 0, 5, 8, 9, 4, 9, 3, 5, 4, 7, 1, 2, 3, 5, 4, 8, 3, 5, 1, 0, 7, 8, 9, 0, 1, 5, 1, 5, 1, 0, 1, 6, 6, 8, 3, 0, 0, 9, 9, 1, 8, 3, 6, 0, 1, 6, 7, 3, 1, 8, 1, 4, 5, 2, 5, 1, 6, 8, 7, 4, 8, 9, 2, 1, 4, 3, 2, 5, 9, 0, 7, 9
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OFFSET
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1,2
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COMMENTS
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For many choices of a,b,c, there are exactly two numbers x having a*x^2 + b*sin(x) = c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
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 A198866, take f(x,u,v) = x^2 + u*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.
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LINKS
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EXAMPLE
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negative: -1.40962400400259624923559397058949354...
positive: 0.63673265080528201088799090383828005...
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MATHEMATICA
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(* Program 1: this sequence and A198867 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.41, -1.40}, WorkingPrecision -> 110]
RealDigits[r] (* this sequence *)
r = x /. FindRoot[f[x] == g[x], {x, .63, .64}, WorkingPrecision -> 110]
(* Program 2: implicit surface of x^2+u*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 6}, {v, u, 12}];
ListPlot3D[Flatten[t, 1]] (* for this sequence *)
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PROG
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(PARI) a=1; b=1; c=1; solve(x=-2, 0, a*x^2 + b*sin(x) - c) \\ G. C. Greubel, Feb 20 2019
(Sage) a=1; b=1; c=1; (a*x^2 + b*sin(x)==c).find_root(-2, 0, x) # G. C. Greubel, Feb 20 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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