OFFSET
1,2
COMMENTS
For some choices of a, b, c, there is a unique value of x satisfying a*x^2+b*x+c=e^x, for other choices, there are two solutions, and for others, three. Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 2.... A201741
1.... 0.... 3.... A201742
1.... 0.... 4.... A201743
1.... 0.... 5.... A201744
1.... 0.... 6.... A201745
1.... 0.... 7.... A201746
1.... 0.... 8.... A201747
1.... 0.... 9.... A201748
1.... 0.... 10... A201749
-1... 0.... 1.... A201750, (x=0)
1.... 1.... 0.... A201769
1.... 1.... 1.... ..(x=0), A201770
1.... 1.... 2.... A201396
1.... 1.... 3.... A201562
1.... 1.... 4.... A201772
1.... 1.... 5.... A201889
1.... 2.... 1.... ..(x=0), A201890
1.... 2.... 2.... A201891
1.... 2.... 3.... A201892
1.... 2.... 4.... A201893
1.... 2.... 5.... A201894
1.... 3.... 3.... A201900
1.... 3.... 4.... A201901
1.... 3.... 5.... A201902
1.... 4.... 5.... A201930
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 A201741, take f(x,u,v)=u*x^2+v-e^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.
EXAMPLE
x=1.31907367685736535441789910952084846442196...
MATHEMATICA
(* Program 1: A201741 *)
a = 1; b = 0; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r] (* A201741 *)
(* Program 2: implicit surface of u*x^2+v=E^x *)
f[{x_, u_, v_}] := u*x^2 + v - E^x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 5}]},
{v, 1, 3}, {u, 0, 5}];
ListPlot3D[Flatten[t, 1]] (* for A201741 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 04 2011
STATUS
approved