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A200338
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Decimal expansion of least x > 0 satisfying x^2 + 1 = tan(x).
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159
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1, 1, 7, 2, 0, 9, 3, 6, 1, 7, 2, 8, 5, 6, 6, 9, 0, 3, 9, 6, 8, 7, 8, 1, 8, 7, 9, 5, 8, 1, 0, 8, 9, 8, 8, 0, 4, 0, 2, 4, 2, 4, 5, 7, 0, 8, 8, 0, 2, 7, 6, 3, 7, 1, 7, 6, 0, 1, 8, 6, 6, 3, 6, 7, 1, 2, 1, 8, 6, 6, 3, 4, 6, 0, 7, 6, 4, 1, 2, 2, 8, 3, 6, 5, 4, 5, 6, 1, 1, 2, 2, 8, 6, 7, 2, 3, 0, 3, 2
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OFFSET
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1,3
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COMMENTS
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For many choices of a,b,c, there is exactly one x satisfying a*x^2 + b*x + c = tan(x) and 0 < x < Pi/2.
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 A200338, take f(x,u,v) = x^2 + u*x + 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.
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LINKS
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EXAMPLE
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x=1.17209361728566903968781879581089880...
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MATHEMATICA
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a = 1; b = 0; c = 1;
f[x_] := a*x^2 + b*x + c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
(* Program 2: implicit surface of x^2+u*x+v=tan(x) *)
f[{x_, u_, v_}] := x^2 + u*x + v - Tan[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.57}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
ListPlot3D[Flatten[t, 1]] (* for A200388 *)
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PROG
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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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