OFFSET
1,2
COMMENTS
For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 1.... A125578
1.... 0.... 2.... A197806
1.... 0.... 3.... A197807
1.... 0.... 4.... A197808
2.... 0.... 1.... A198110
2.... 0.... 3.... A198111
3.... 0.... 1.... A198211
3.... 0.... 2.... A198212
3.... 0.... 4.... A198213
4.... 0.... 1.... A198347
4.... 0.... 3.... A198348
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 A197737, take f(x,u,v)=x^2+u*x-v*cos(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
negative: -1.25115183522076481159287006878816185994...
positive: 0.55000934992726156666495361947172926116...
MATHEMATICA
(* Program 1: A197738 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2, 1}]
r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]
RealDigits[r1] (* A197737 *)
r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]
RealDigits[r1] (* A197738 *)
(* Program 2: implicit surface of x^2+u*x=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]] (* for A197737 *)
PROG
(PARI) A197737_vec(N=150)={localprec(N+10); digits(solve(x=-1.5, -1, x^2+x-cos(x))\.1^N)} \\ M. F. Hasler, Aug 05 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 20 2011
STATUS
approved