%I
%S 1,1,5,0,5,8,4,9,6,7,4,1,8,6,6,3,9,4,9,5,3,4,9,3,3,7,3,3,6,1,3,7,8,8,
%T 1,9,5,7,6,6,8,3,7,4,9,4,8,4,4,2,4,2,3,4,1,1,8,3,3,9,2,5,1,8,0,8,8,3,
%U 2,2,5,4,6,1,7,6,4,1,1,7,2,8,0,1,3,6,7,5,4,4,1,4,5,2,4,6,9,9,5
%N Decimal expansion of x>0 satisfying x*cosh(x)=2.
%C For many choices of u and v, there is exactly one x>0 satisfying x*cosh(u*x)=v. Guide to related sequences, with graphs included in Mathematica programs:
%C u.... v.... x
%C 1.... 1.... A069814
%C 1.... 2.... A201939
%C 1.... 3.... A201943
%C 2.... 1.... A201944
%C 3.... 1.... A201945
%C 2.... 2.... A202283
%C 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.
%C For an example related to A201939, take f(x,u,v)=x*cosh(u*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 singlevalued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
%e x=1.15058496741866394953493373361378819576...
%t (* Program 1: A201939 *)
%t u = 1; v = 2;
%t f[x_] := x*Cosh[u*x]; g[x_] := v
%t Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin > {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision > 110]
%t RealDigits[r] (* A201939 *)
%t (* Program 2: implicit surface of u*cosh(x)=v *)
%t f[{x_, u_, v_}] := x*Cosh[u*x]  v;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 20}, {u, 1, 9}];
%t ListPlot3D[Flatten[t, 1]] (* for A201939 *)
%Y Cf. A201946.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Dec 15 2011
