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%I #6 Mar 30 2012 18:58:03
%S 1,2,4,9,3,9,4,3,3,6,6,4,6,3,2,4,4,7,2,5,1,1,2,7,4,3,2,1,2,6,1,0,0,8,
%T 1,2,3,4,6,9,4,4,1,3,0,0,9,0,1,5,2,9,6,9,6,2,9,7,2,6,0,7,6,8,6,8,8,2,
%U 1,2,3,9,7,3,1,0,6,2,2,9,8,3,0,0,5,3,2,5,7,7,0,8,2,8,6,5,3,9,1
%N Decimal expansion of x>0 satisfying x*sinh(x)=2.
%C For many choices of u and v, there is exactly one x>0 satisfying x*sinh(u*x)=v. Guide to related sequences, with graphs included in Mathematica programs:
%C u.... v.... x
%C 1.... 1.... A133867
%C 1.... 2.... A201946
%C 1.... 3.... A202243
%C 2.... 1.... A202244
%C 3.... 1.... A202245
%C 2.... 2.... A202284
%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 A199597, take f(x,u,v)=x*sinh(ux)-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.
%e x=1.2493943366463244725112743212610081234694...
%t (* Program 1: A201946 *)
%t u = 1; v = 2;
%t f[x_] := x*Sinh[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, 1.2, 1.3}, WorkingPrecision -> 110]
%t RealDigits[r] (* A201946 *)
%t (* Program 2: implicit surface of u*sinh(x)=v *)
%t f[{x_, u_, v_}] := x*Sinh[u*x] - v;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
%t ListPlot3D[Flatten[t, 1]] (* for A201946 *)
%Y Cf. A201939.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Dec 15 2011
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