OFFSET
0,1
COMMENTS
For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... least x, greatest 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 A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
EXAMPLE
least x: 0.659266045766946074537348579563067611...
greatest x: 1.2710268008159460640047188480978502...
MATHEMATICA
(* Program 1: A199949 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r] (* A199949 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
RealDigits[r] (* A199950 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
ListPlot3D[Flatten[t, 1]] (* for A199949 *)
PROG
(PARI) a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jul 05 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 12 2011
EXTENSIONS
A-number corrected by Jaroslav Krizek, Nov 27 2011
STATUS
approved