

A201946


Decimal expansion of x>0 satisfying x*sinh(x)=2.


6



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, 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, 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
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OFFSET

1,2


COMMENTS

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:
u.... v.... x
1.... 1.... A133867
1.... 2.... A201946
1.... 3.... A202243
2.... 1.... A202244
3.... 1.... A202245
2.... 2.... A202284
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 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 singlevalued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.


LINKS

Table of n, a(n) for n=1..99.


EXAMPLE

x=1.2493943366463244725112743212610081234694...


MATHEMATICA

(* Program 1: A201946 *)
u = 1; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin > {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision > 110]
RealDigits[r] (* A201946 *)
(* Program 2: implicit surface of u*sinh(x)=v *)
f[{x_, u_, v_}] := x*Sinh[u*x]  v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
ListPlot3D[Flatten[t, 1]] (* for A201946 *)


CROSSREFS

Cf. A201939.
Sequence in context: A081344 A227272 A021405 * A341352 A301514 A269063
Adjacent sequences: A201943 A201944 A201945 * A201947 A201948 A201949


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Dec 15 2011


STATUS

approved



