A201939 - OEIS

%I #9 Feb 04 2025 10:05:45

%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 single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e 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